Characterizations of Recently Introduced Univariate Continuous Distributions III 153619297X, 9781536192971

Citation preview

MATHEMATICS RESEARCH DEVELOPMENTS

CHARACTERIZATIONS OF RECENTLY INTRODUCED UNIVARIATE CONTINUOUS DISTRIBUTIONS III

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

MATHEMATICS RESEARCH DEVELOPMENTS Additional books and e-books in this series can be found on Nova’s website under the Series tab.

MATHEMATICS RESEARCH DEVELOPMENTS

CHARACTERIZATIONS OF RECENTLY INTRODUCED UNIVARIATE CONTINUOUS DISTRIBUTIONS III

G. G. HAMEDANI

Copyright © 2021 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data

ISBN: H%RRN

Published by Nova Science Publishers, Inc. † New York

Contents Preface 1 Introduction 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Extended Weighted Exponential (EWE) . . . . . . . . . . . . . . . 1.1.2 Exponentiated Generalized Extended Pareto (EGEP) . . . . . . . . 1.1.3 Inverse Weibull Generator (IWG) . . . . . . . . . . . . . . . . . . 1.1.4 Generalized Burr X-G (GBX-G) . . . . . . . . . . . . . . . . . . . 1.1.5 Generalized Lindley Power Series (GLPS) . . . . . . . . . . . . . 1.1.6 Transmuted Exponentiated Additive Weibull (TEAW) . . . . . . . 1.1.7 Exponentiated Weighted Exponential (EWE) . . . . . . . . . . . . 1.1.8 New Weighted Exponential (NWE) . . . . . . . . . . . . . . . . . 1.1.9 Gompertz Lomax (GoLom) . . . . . . . . . . . . . . . . . . . . . 1.1.10 Chen’s Two-Parameter Exponential Power Life-Testing (CTPEPLT) 1.1.11 Inverted Weighted Exponential (IWE) . . . . . . . . . . . . . . . . 1.1.12 Kumaraswamy Marshall-Oklin Exponential (KMOE) . . . . . . . . 1.1.13 Kumaraswamy Weibull (KumW) . . . . . . . . . . . . . . . . . . . 1.1.14 Kumaraswamy Half-Logistics (KH-L) . . . . . . . . . . . . . . . . 1.1.15 Transmuted Generalized Linear Exponential (TGLE) . . . . . . . . 1.1.16 Kumaraswamy-Chen (Kw-Chen), Kumaraswamy-XTG (Kw-XTG) and Kumaraswamy Flexible Weibull (Kw-FW) . . . . . 1.1.17 Exponentiated Generalized Inverted Exponential (EGIE) . . . . . . 1.1.18 Exponentiated Generalized Exponentiated Exponential (EGEE) . . 1.1.19 Transmuted Exponentiated U-quadratic (TEUq) . . . . . . . . . . . 1.1.20 Generalized Gamma Burr III (GGBIII) . . . . . . . . . . . . . . . 1.1.21 Beta Skew-t (BST) . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.22 A Class of Lindley and Weibull (ACLW) . . . . . . . . . . . . . . 1.1.23 Generalized Inverted Kumaraswamy (GIKum) . . . . . . . . . . . 1.1.24 Exponentiated Generalized Extended Pareto (EGEP) . . . . . . . . 1.1.25 Benktander Type II (BType II) . . . . . . . . . . . . . . . . . . . . 1.1.26 Generalized Transmuted Fréchet (GTFr) . . . . . . . . . . . . . . . 1.1.27 Beta Linear Failure Rate Power Series (BLFRPS) . . . . . . . . . . 1.1.28 Odd Lomax-G (OLxG) . . . . . . . . . . . . . . . . . . . . . . . . 1.1.29 Exponentiated Power Generalized Weibull (EPGW) . . . . . . . .

xxiii 1 16 16 16 17 17 18 18 19 19 20 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 27 28 28 29

Contents

vi 1.1.30 1.1.31 1.1.32 1.1.33 1.1.34 1.1.35 1.1.36 1.1.37 1.1.38 1.1.39 1.1.40 1.1.41 1.1.42 1.1.43 1.1.44 1.1.45 1.1.46 1.1.47 1.1.48 1.1.49 1.1.50 1.1.51 1.1.52 1.1.53 1.1.54 1.1.55 1.1.56 1.1.57 1.1.58 1.1.59 1.1.60 1.1.61 1.1.62 1.1.63 1.1.64 1.1.65 1.1.66 1.1.67 1.1.68 1.1.69 1.1.70 1.1.71 1.1.72 1.1.73 1.1.74

Lindley Weibull (LiW) . . . . . . . . . . . . . . . . . . . . . . Marshall-Olkin Alpha Power (MOAP) . . . . . . . . . . . . . . Zero Spiked Gamma Weibull (ZSGW) . . . . . . . . . . . . . . Inverted Nadarajah-Haghighi (INH) . . . . . . . . . . . . . . . Marshall-Olkin Generated Gamma (MOGG) . . . . . . . . . . Gamma Generalized Normal (GGN) . . . . . . . . . . . . . . . Transmuted Transmuted-G (TTG) . . . . . . . . . . . . . . . . Power Binomial Exponential 2 (PBE2) . . . . . . . . . . . . . Beta Burr III (BBIII) . . . . . . . . . . . . . . . . . . . . . . . Muth Generated (MG) . . . . . . . . . . . . . . . . . . . . . . Weibull-Lindley (WLn) . . . . . . . . . . . . . . . . . . . . . Weibull-G Power Series (WGPS) . . . . . . . . . . . . . . . . Three Parameter Generalized Lindley (TPGL) . . . . . . . . . . Odd Lindley Exponentiated Weibull (OLEW) . . . . . . . . . . Extended Odd Fréchet-G (EOF-G) . . . . . . . . . . . . . . . . Alpha Power Transformation Poisson Lindley (APTPL) . . . . Alpha Logarithm Transmuted Fréchet (ALTF) . . . . . . . . . . Burr-Weibull Power Series (BWPS) . . . . . . . . . . . . . . . Zografos-Balakrishnan Fréchet (ZBFr) . . . . . . . . . . . . . Cubic Transmuted Weibull (CTW) . . . . . . . . . . . . . . . . Cubic Rank Transmuted Kumaraswamy (CRTKw) . . . . . . . Cubic Transmuted Weibull (CTW) . . . . . . . . . . . . . . . . Type II Kumaraswamy Half Logistic-Generated (TIIKwHL-G) . Odd Log-Logistic Generalized Inverse Gaussian (OLLGIG) . . General Class (GC) . . . . . . . . . . . . . . . . . . . . . . . . Poisson Burr Type X Log-Logistic (PBXLL) . . . . . . . . . . New Odd Generalized Exponential-Exponential (NOGE-E) . . Kumaraswamy Extension Exponential (KEE) . . . . . . . . . . Extended Enlarg Transmuted Exponential (EETE) . . . . . . . Marshall-Olkin Extended Power Function (MOEPF) . . . . . . Generalized Odd Log-Logistic Exponential (GOLLE) . . . . . Exponentiated Transmuted Power Function (ETPF) . . . . . . . Type II Half Logistic Weibull (TIIHLW) . . . . . . . . . . . . . Exponentiated Generalized Inverse Rayleigh (EGIR) . . . . . . Generalized Transmuted Gompertz-Makeham (GTGM) . . . . Jamal Weibull-X (JW-X) . . . . . . . . . . . . . . . . . . . . . Nasir Logistic-X (NL-X) . . . . . . . . . . . . . . . . . . . . . Jamal Logistics-X (JL-X) . . . . . . . . . . . . . . . . . . . . Nasir Weibull-Generalized (NW-G) . . . . . . . . . . . . . . . Weibull Pareto (WP) . . . . . . . . . . . . . . . . . . . . . . . Slashed Power-Lindley (SPL) . . . . . . . . . . . . . . . . . . Exponentiated Generalized Pareto (EGP) . . . . . . . . . . . . Inverse Weighted Lindley (IWL) . . . . . . . . . . . . . . . . . Unit-Inverse Gaussian (UIG) . . . . . . . . . . . . . . . . . . . Weibull-Moment Exponential (WME) . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 30 30 31 31 32 32 33 33 33 34 34 34 35 35 36 36 37 37 38 38 38 39 39 40 40 40 41 41 42 42 43 43 44 44 45 45 45 46 46 46 47 47 47 48

Contents 1.1.75 1.1.76 1.1.77 1.1.78 1.1.79 1.1.80 1.1.81 1.1.82 1.1.83 1.1.84 1.1.85 1.1.86 1.1.87

Generalized Odd Burr III-G (GOBIII-G) . . . . . . . . . . . . . Generalized Odd Fréchet-G (GOFr-G) . . . . . . . . . . . . . . . Type I Half Logistic Power Lindley (TIHLPL) . . . . . . . . . . New Alpha-Power Transformation (NAPT) . . . . . . . . . . . . Functional Weighted Exponential (FWE) . . . . . . . . . . . . . Odd Burr III G-Negative Binomial (OBIIIGNB) . . . . . . . . . Type I Half-Logistic Exponential (TIHLE) . . . . . . . . . . . . Unit-Marshall-Olkin Extended Exponential (UMOEE) . . . . . . Generalization of Two-Parameter Lindley (GTPL) . . . . . . . . Odd Lindley Fréchet (OLiFr) . . . . . . . . . . . . . . . . . . . . Topp-Leone Mukherjee-Islam (TLMI) . . . . . . . . . . . . . . . Weibull-Lomax (WL) . . . . . . . . . . . . . . . . . . . . . . . Zero Truncated Poisson Topp-Leone Exponentiated Weibull (ZTPTLEW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.88 Zero Truncated Poisson Topp-Leone Burr XII (ZTPTLBXII) . . . 1.1.89 General Transmuted Family (GTF) . . . . . . . . . . . . . . . . 1.1.90 Poisson Topp-Leone Inverse Weibull (PTLIW) . . . . . . . . . . 1.1.91 Odd Burr-G Poisson (OBGP) . . . . . . . . . . . . . . . . . . . 1.1.92 Exponentiated Mukherjee-Islam (EMI) . . . . . . . . . . . . . . 1.1.93 Generalized Transmuted Power Function (GTPF) . . . . . . . . . 1.1.94 Poisson Exponentiated Erlang-Truncated Exponential (PEETE) . 1.1.95 Minimum Guarantee Lindley (MGL) . . . . . . . . . . . . . . . 1.1.96 Inverted Beta (IB) . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.97 Cubic Transmuted Pareto (CTP) . . . . . . . . . . . . . . . . . . 1.1.98 Zero Truncated Poisson Topp Leone Weibull (ZTPTLW) . . . . . 1.1.99 A New (AN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.100 Exponentiated Kumarsawamy-G (EKw-G) . . . . . . . . . . . . 1.1.101 Log-Sinh Cauchy Promotion (LSCp) . . . . . . . . . . . . . . . 1.1.102 Modified Beta Modified-Weibull (MBMW) . . . . . . . . . . . . 1.1.103 Kumaraswamy Generalized Linear Exponential (Kw-GLE) . . . . 1.1.104 Weighted Inverse Gamma (WIG) . . . . . . . . . . . . . . . . . 1.1.105 Odd Log-Logistic Generalized Half-Normal Poisson (OLLGHNP) 1.1.106 Topp-Leone Weighted Weibull (TLWW) . . . . . . . . . . . . . . 1.1.107 Burr-Hatke-G (BH-G) . . . . . . . . . . . . . . . . . . . . . . . 1.1.108 Burr-Hatke Exponentiated Weibull (BHEW) . . . . . . . . . . . 1.1.109 Generalized Log-Lindley (GLL) . . . . . . . . . . . . . . . . . . 1.1.110 Weibull Generalized Log-Logistic (WGLL) . . . . . . . . . . . . 1.1.111 Transmuted Generalized Odd Generalized Exponential-G (TGOGE-G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.112 New Three Parameter Paralogistic (NTPL) . . . . . . . . . . . . 1.1.113 Janardan-Power Series (JPS) . . . . . . . . . . . . . . . . . . . . 1.1.114 Exponentiated Generalized Extended Gompertz (EGEG) . . . . . 1.1.115 Power-Exponential Hazard Rate (P-EHR) . . . . . . . . . . . . . 1.1.116 Inverse Power Lomax (IPL) . . . . . . . . . . . . . . . . . . . . 1.1.117 Exponentiated Kumaraswamy-Weibull (EK-W) . . . . . . . . . .

vii . . . . . . . . . . . .

48 49 49 50 50 51 51 51 52 52 52 53

. . . . . . . . . . . . . . . . . . . . . . . .

53 53 54 54 54 55 55 56 56 57 57 57 58 58 59 59 59 60 60 61 61 62 62 62

. . . . . . .

63 63 64 64 64 65 65

viii

Contents 1.1.118 Power Function Power Series (PFPS) . . . . . . . . . . . . . . 1.1.119 Exponentiated Negative Binomial (ENB) . . . . . . . . . . . . 1.1.120 Burr-Hatke Exponential (BHE) . . . . . . . . . . . . . . . . . 1.1.121 X Gamma Weibull (XGW) . . . . . . . . . . . . . . . . . . . . 1.1.122 Exponentiated Exponential Logistic (EEL) . . . . . . . . . . . 1.1.123 Reduced New Modified Weibull (RNMW) . . . . . . . . . . . 1.1.124 Extended Weibull-G (EW-G) . . . . . . . . . . . . . . . . . . . 1.1.125 Composite Generalizers of Weibull (CGW) . . . . . . . . . . . 1.1.126 Odd Log-Logistic Exponentiated Gumbel (OLLEGu) . . . . . . 1.1.127 Harris Extended Lindley (HEL) . . . . . . . . . . . . . . . . . 1.1.128 Odd Burr III Weibull (OBIIIW) . . . . . . . . . . . . . . . . . 1.1.129 Power Lindley Generated (PLG) . . . . . . . . . . . . . . . . . 1.1.130 Inverse Gompertz (IG) . . . . . . . . . . . . . . . . . . . . . . 1.1.131 Hyperbolic Sine Rayleigh (HS-R) . . . . . . . . . . . . . . . . 1.1.132 Type II Topp-Leone Generated (TIITL-G) . . . . . . . . . . . . 1.1.133 Exponentiated Inverse Rayleigh (EIR) . . . . . . . . . . . . . . 1.1.134 Exponentiated New Weighted Weibull (ENWW) . . . . . . . . 1.1.135 Beta Transmuted Weighted Exponential (BTWE) . . . . . . . . 1.1.136 Odd Lindley Exponentiated Weibull (OLi-EW) . . . . . . . . . 1.1.137 Modified Weibull-G (MW-G) . . . . . . . . . . . . . . . . . . 1.1.138 Compound Gamma and Lindley (GaL) . . . . . . . . . . . . . 1.1.139 Odd Weibull (OW) . . . . . . . . . . . . . . . . . . . . . . . . 1.1.140 Topp-Leone Generalized Inverted Kumarswamy (TLGIKw) . . 1.1.141 Marshall-Olkin Burr X (MOBX) . . . . . . . . . . . . . . . . . 1.1.142 New Extended Generalized Burr III (NEGBIII) . . . . . . . . . 1.1.143 Generalized Inverse Weibull-Generalized Inverse Weibull (GIW-GIW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.144 Exponentiated Burr XII Power Series (EBXIIPS) . . . . . . . . 1.1.145 Burr XII Weibull Logarithmic (BWL) . . . . . . . . . . . . . . 1.1.146 Odd Log-Logistic Geometric Normal (OLLGN) . . . . . . . . 1.1.147 Marshall-Olkin Extended Flexible Weibull (MOEFW) . . . . . 1.1.148 Topp-Leone Inverse Weibull (TLIW) . . . . . . . . . . . . . . 1.1.149 Odd Log-Logistic Marshall-Olkin Power Lindley (OLLMOPL) 1.1.150 Topp-Leone Lomax (TLLo) . . . . . . . . . . . . . . . . . . . 1.1.151 Exponentiated Topp-Leone (ETL) . . . . . . . . . . . . . . . . 1.1.152 Topp-Leone Generator (TLG) . . . . . . . . . . . . . . . . . . 1.1.153 Topp-Leone Generated q-Exponential (TLG-qE) . . . . . . . . 1.1.154 Odd Hyperbolic Cosine KG (OHC-KG) . . . . . . . . . . . . . 1.1.155 Generalized Gudermannian (GG) . . . . . . . . . . . . . . . . 1.1.156 New Extended Alpha Power Transformed (NEAPT) . . . . . . 1.1.157 Exponentiated Odd Log-Logistic-G (EOLL-G) . . . . . . . . . 1.1.158 Zero Truncated Poisson Topp Leone Weibull (ZTPTLW) . . . . 1.1.159 Type II Generalized Topp-Leone-G (TIIGTL-G) . . . . . . . . 1.1.160 Burr XII Exponentiated Weibull (BXIIEW) . . . . . . . . . . . 1.1.161 Generalized Odd Log-Logistics Inverse Weibull (GOLLIW) . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

66 66 67 67 67 68 68 68 69 70 70 71 71 72 72 72 73 73 74 74 75 75 76 76 77

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

77 78 78 78 79 79 80 80 81 81 82 82 82 83 83 83 84 85 85

Contents 1.1.162 Centered Skew-Normal Birnbaum-Saunders (CSNBS) . . . . . . . 1.1.163 Logarithmic Kumarswamy (LKu) . . . . . . . . . . . . . . . . . . 1.1.164 Marshall-Olkin Odd Lindley G (MOOL-G) . . . . . . . . . . . . . 1.1.165 Generalized Transmuted Fréchet (GTF) . . . . . . . . . . . . . . . 1.1.166 Weibull Generalized Exponentiated Weibull (WGEW) . . . . . . . 1.1.167 Odd Log-Logistic Poisson-G (OLLP-G) . . . . . . . . . . . . . . . 1.1.168 New Lindley Exponential (NLE) . . . . . . . . . . . . . . . . . . . 1.1.169 Odd Log-Logistic Exponentiated Weibull (OLLEW) . . . . . . . . 1.1.170 Another Odd Log-Logistic Logarithmic (AOLLL-G) . . . . . . . . 1.1.171 Type I Half-Logistic (TIHL-G) . . . . . . . . . . . . . . . . . . . . 1.1.172 Odd Burr- Generalized (OBu-G) . . . . . . . . . . . . . . . . . . . 1.1.173 Odd Log-Logistic Topp-Leone G (OLLTL-G) . . . . . . . . . . . . 1.1.174 Topp-Leone Odd Lindley-G (TLOL-G) . . . . . . . . . . . . . . . 1.1.175 Odd Log-Logistic Log-Normal (OLL-LN) . . . . . . . . . . . . . . 1.1.176 Odd Log-Logistic Generalized Gompertz (OLLGG) . . . . . . . . 1.1.177 New Odd Log-Logistic Half-Logistic (NOLL-HL) . . . . . . . . . 1.1.178 Zografos-Balakrishnan Odd Log-Logistic Generalized HalfNormal (ZNOLL-GHN) . . . . . . . . . . . . . . . . . . . . . . . 1.1.179 Odd Log-Logistic Marshall-Olkin Generalized Half-Normal (OLLMOGHN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.180 Power-Linear Hazard Rate (P-LHR) . . . . . . . . . . . . . . . . . 1.1.181 Alpha-Power Pareto (APP) . . . . . . . . . . . . . . . . . . . . . . 1.1.182 Exponentiated Odd Log-Logistic Normal (EOLLN) . . . . . . . . . 1.1.183 Extended Odd Fréchet-G (EOF-G) . . . . . . . . . . . . . . . . . . 1.1.184 Log-Odd Log-Logisticc Birnbaum-Saunders-Poisson (OLLBSP) . . 1.1.185 Zografos-Balakrishnan Lindley-Poisson (ZB-LP) . . . . . . . . . . 1.1.186 One-Parameter Weibull-Type (1P-Weibull) . . . . . . . . . . . . . 1.1.187 Beta Odd Lindley-G (BOL-G) . . . . . . . . . . . . . . . . . . . . 1.1.188 A Distribution For Instantaneous Failures (ADFIF) . . . . . . . . . 1.1.189 Generalized Inverse Weibull- Generalized Inverse Weibull (GIWGIW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.190 Beta Lindley Geometric (BLGc) . . . . . . . . . . . . . . . . . . . 1.1.191 Odd Log-Logistic Generalized Half-Normal (OLLGHN) . . . . . . 1.1.192 Odd Lindley Lomax (OLLo) . . . . . . . . . . . . . . . . . . . . . 1.1.193 Burr X Exponentiated Lomax (BrXELx) . . . . . . . . . . . . . . 1.1.194 Extended Normal (EN) . . . . . . . . . . . . . . . . . . . . . . . . 1.1.195 Poisson-X (P-X) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.196 α-Power Transmuted Generalized Exponential (αPTGE) . . . . . . 1.1.197 Weibull Exponentiated Exponential (WEE) . . . . . . . . . . . . . 1.1.198 Weighted Modified Weibull (WMW) . . . . . . . . . . . . . . . . 1.1.199 Poisson Nadarajah-Haghighi (PNH) . . . . . . . . . . . . . . . . . 1.1.200 Poisson Burr X Weibull (PBrXW) . . . . . . . . . . . . . . . . . . 1.1.201 Marshall-Olkin Extended Exponential (MOEE) . . . . . . . . . . . 1.1.202 New Generalization of Weibull-Exponential (NGWE) . . . . . . . 1.1.203 Odd Fréchet Inverse Weibull (OFIW) . . . . . . . . . . . . . . . .

ix 86 86 87 87 88 89 89 90 90 91 91 92 92 93 93 94 94 95 95 95 96 96 97 97 97 98 98 98 99 99 100 100 100 101 101 102 102 103 103 104 104 105

x

Contents 1.1.204 Topp-Leone-G Poisson (TL-GP) . . . . . . . . . . . . . . . . . . 1.1.205 Alpha Power Transmuted Extended Exponential (APTEE) . . . . 1.1.206 Odd Fréchet Inverse Rayleigh (OFIR) . . . . . . . . . . . . . . . 1.1.207 Odd Inverse Pareto-G (OIP-G) . . . . . . . . . . . . . . . . . . . 1.1.208 Type I Half-Logistic Burr X (TIHLBX ) . . . . . . . . . . . . . . . 1.1.209 Geometric Lindley Poisson 1 (GLP1) . . . . . . . . . . . . . . . 1.1.210 Odd Fréchet Inverse Exponential (OFIE) . . . . . . . . . . . . . 1.1.211 Truncated Weibull Power Lomax (TWPL) . . . . . . . . . . . . . 1.1.212 Generalized Extended Inverse Weibull (GEIW) . . . . . . . . . . 1.1.213 Lomax-Lindley (L-L) . . . . . . . . . . . . . . . . . . . . . . . . 1.1.214 Generalized Gompertz-Generalized Gompertz (GG-GG) . . . . . 1.1.215 Exponentiated Weibull Weibull (EWW) . . . . . . . . . . . . . . 1.1.216 Weibull-Inverse Lomax (WIL) . . . . . . . . . . . . . . . . . . . 1.1.217 Burr XII Inverse Rayleigh (BXII-IR) . . . . . . . . . . . . . . . 1.1.218 Kumarswamy Type I Half Logistic (KwTIHL-G) . . . . . . . . . 1.1.219 Odd Generalized Exponential Power Function (OGEPF) . . . . . 1.1.220 Generalized Odd Lomax Generated (GOLG) . . . . . . . . . . . 1.1.221 Inverse Weibull Geometric (IWG) . . . . . . . . . . . . . . . . . 1.1.222 Inverse Weibull Poisson (IWP) . . . . . . . . . . . . . . . . . . . 1.1.223 Transmuted Four Parameters Generalized Log-Logistic (TFPGLL) 1.1.224 Burr X-Kumaraswamy (BXKw) . . . . . . . . . . . . . . . . . . 1.1.225 Transmuted Generalized Gamma (TGG) . . . . . . . . . . . . . . 1.1.226 Generalized Extended Exponential-Weibull (GEEW) . . . . . . . 1.1.227 Nadarajah-Haghigh Geometric (NHG) . . . . . . . . . . . . . . . 1.1.228 Nadarajah-Haghighi Lindley (NHL) . . . . . . . . . . . . . . . . 1.1.229 Exponentiated Log-Sinh Cauchy (ELSC) . . . . . . . . . . . . . 1.1.230 Modified Fréchet (MFr) . . . . . . . . . . . . . . . . . . . . . . 1.1.231 Cubic Transmuted Uniform (CTU) . . . . . . . . . . . . . . . . . 1.1.232 Beta-G Poisson (BGP) . . . . . . . . . . . . . . . . . . . . . . . 1.1.233 Cosine-Sine Transformation (CST) . . . . . . . . . . . . . . . . 1.1.234 Generalized Burr XII Power Series (GBXIIPS) . . . . . . . . . . 1.1.235 Transmuted Extended Exponential (TEE) . . . . . . . . . . . . . 1.1.236 Exponentiated Generalized Power Series (EGPS) . . . . . . . . . 1.1.237 Marshall-Olkin Alpha Power Inverse Exponential (MOAPIE) . . 1.1.238 Transmuted Topp Leone Exponentiated Fréchet (TTLEFr) . . . . 1.1.239 Generalized Gudermannian (GG) . . . . . . . . . . . . . . . . . 1.1.240 Alpha Skew Generalized Gudermannian (ASGG) . . . . . . . . . 1.1.241 Truncated-Logistic Skew-Symmetric (TLSS) . . . . . . . . . . . 1.1.242 Combined Exponential-Normal {Generalized Weibull} (CE-N{GW}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.243 Type II Half Logistic Ibrahim (TIIHLI) . . . . . . . . . . . . . . 1.1.244 Modified Generalized Marshall-Olkin (MGMO) . . . . . . . . . 1.1.245 Transmuted Extended Lomax (TEL) . . . . . . . . . . . . . . . . 1.1.246 Topp-Leone Odd Log-Logistic Exponential (TLOLLEx) . . . . . 1.1.247 Odd Birnbaum-Saunders (OBS) . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 106 106 107 107 108 108 109 109 109 110 110 111 111 112 112 113 113 113 114 114 115 115 115 116 116 116 116 117 117 118 118 118 119 120 120 120

. . . . . .

120 121 121 122 122 123

Contents 1.1.248 Gamma Dual Weibull (Γ2 -W) . . . . . . . . . . . . . . . . . . 1.1.249 Quasi XGamma-Poisson (QXGP) . . . . . . . . . . . . . . . . 1.1.250 Modified Extended Generalized Exponential (MEGE) . . . . . 1.1.251 Exponentiated Nadarajah Haghighi Poisson (ENHP) . . . . . . 1.1.252 Extended New Generalized Exponential (ENGE) . . . . . . . . 1.1.253 Transmuted Singh-Maddala (TSM) . . . . . . . . . . . . . . . 1.1.254 Transmuted Half Logistic (THL) . . . . . . . . . . . . . . . . . 1.1.255 Kumarswamy Exponentiated U-Quadratic (KwEUQ) . . . . . . 1.1.256 New Generalized Transmuted Inverse Exponential (NGT-IE) . . 1.1.257 Transmuted Lomax Exponential (TLE) . . . . . . . . . . . . . 1.1.258 Extended Pranav (EP) . . . . . . . . . . . . . . . . . . . . . . 1.1.259 Generalized Odd Lindley-G (GOLi-G) . . . . . . . . . . . . . 1.1.260 Leaned Normal (LN) . . . . . . . . . . . . . . . . . . . . . . . 1.1.261 XGamma (XG) . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.262 Quasi XGamma-Geometric (QXGGc) . . . . . . . . . . . . . . 1.1.263 Marshall-Olkin Modified Burr III (MOMBIII) . . . . . . . . . . 1.1.264 Type II Exponentiated Half Logistic Generated (TIIEHL-G) . . 1.1.265 Weighted XGamma (WXG) . . . . . . . . . . . . . . . . . . . 1.1.266 Length Biased XGamma (LBXG) . . . . . . . . . . . . . . . . 1.1.267 Wrapped XGamma (WRXG) . . . . . . . . . . . . . . . . . . 1.1.268 Linearly Decreasing Stress Weibull (LDSWeibull) . . . . . . . 1.1.269 L-Logistic (L-Logistic) . . . . . . . . . . . . . . . . . . . . . . 1.1.270 Normal Generalized Hyperbolic Secant (NGHS) . . . . . . . . 1.1.271 Generalized Inverse Pareto-G (GIP-G) . . . . . . . . . . . . . . 1.1.272 Kumarsawamy Marshall-Olkin Modified Weibull (KMOMW) . 1.1.273 Wrapped Lindley-Exponential (WRLE) . . . . . . . . . . . . . 1.1.274 Odoma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.275 Wibull Inverse Lomax (WIL) . . . . . . . . . . . . . . . . . . . 1.1.276 Poisson Burr X Pareto Type II (PBXPTII) . . . . . . . . . . . . 1.1.277 Burr XII Fréchet (BrXIIFr) . . . . . . . . . . . . . . . . . . . . 1.1.278 Burr X Fréchet (BrXFr) . . . . . . . . . . . . . . . . . . . . . 1.1.279 Odd Generalized Exponential Type-I Generalized Half Logistic (OGET-IGHL) . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.280 Burr X Nadarajah Haghighi (BXNH) . . . . . . . . . . . . . . 1.1.281 Topp-Leone Compound Rayleigh (TLCR) . . . . . . . . . . . . 1.1.282 Dual Exponentiated Weibull (DEW) . . . . . . . . . . . . . . . 1.1.283 unit-Improved Second-Degree Lindley (unit-ISDL) . . . . . . . 1.1.284 Kumaraswamy Odd Lindley-G (KOL-G) . . . . . . . . . . . . 1.1.285 Transmuted Type I Generalized Logistic (TTIGL) . . . . . . . . 1.1.286 Truncated Discrete Linnik Weibull (TDLW) . . . . . . . . . . . 1.1.287 Raised Cosine (RC) . . . . . . . . . . . . . . . . . . . . . . . 1.1.288 Transmuted Exponentiated Weibull (TEW) . . . . . . . . . . . 1.1.289 Generalized Marshall-Olkin Extended Burr-III (GMOBIII) . . . 1.1.290 Modified Beta Linear Exponential (MBLE) . . . . . . . . . . . 1.1.291 Zero-Truncated Poisson-Power Function (ZTPPF) . . . . . . .

xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 124 124 125 126 126 127 127 128 128 129 129 130 130 131 131 132 132 133 133 133 134 134 134 135 135 135 136 136 137

. . . . . . . . . . . . .

. . . . . . . . . . . . .

137 137 138 138 139 139 139 140 140 140 141 141 142

xii

Contents 1.1.292 Cubic Transmuted Pareto (CTP) . . . . . . . . . . . . . . . 1.1.293 Inverse XGamma (IXG) . . . . . . . . . . . . . . . . . . . 1.1.294 Alpha-Power Transformed Lindley (αPTL) . . . . . . . . . 1.1.295 Log-Odd Logistic-Weibull (LOLW) . . . . . . . . . . . . . 1.1.296 Skewed Generalized Logistic (SGL) . . . . . . . . . . . . . 1.1.297 Weighted T-X (WTX) . . . . . . . . . . . . . . . . . . . . 1.1.298 Zero-Truncated Poisson Exponentiated Gamma (ZTPEG) . 1.1.299 Power Gompertz (PG) . . . . . . . . . . . . . . . . . . . . 1.1.300 Transmuted New Weibull Pareto (TNWP) . . . . . . . . . . 1.1.301 Lomax Weibull (LoW) . . . . . . . . . . . . . . . . . . . . 1.1.302 Odd Log-Logistic Geometric-G (OLLG-G) . . . . . . . . . 1.1.303 Marshall-Olkin Extended Quasi Lindley (MOEQL) . . . . . 1.1.304 Poisson Rayleigh Log-Logistic (PRLL) . . . . . . . . . . . 1.1.305 Generalized Marshall-Olkin Extended Burr XII (GMOBXII) 1.1.306 Type I New Heavy Tailed Weibull (TINHT-W) . . . . . . . 1.1.307 Arcsine Weibull (AS-W) . . . . . . . . . . . . . . . . . . . 1.1.308 Truncated Inverted Kumaraswamy Generated (TIK-G) . . . 1.1.309 Lomax Gompertz-Makeham (LOGOMA) . . . . . . . . . . 1.1.310 Exponential Transmuted Fréchet (ETF) . . . . . . . . . . . 1.1.311 Marshall-Olkin Extended Weibull Exponential (MOEWE) . 1.1.312 Extended Log-Logistic (ELL) . . . . . . . . . . . . . . . . 1.1.313 Log-Odd Normal Generalized (LONG) . . . . . . . . . . . 1.1.314 Marshall-Olkin Generalized Burr XII (MOGBXII) . . . . . 1.1.315 Generalized Inverse Lindley (GIL) . . . . . . . . . . . . . . 1.1.316 Normal-C (N-C) . . . . . . . . . . . . . . . . . . . . . . . 1.1.317 Extended Power Lindley-G (EPL-G) . . . . . . . . . . . . . 1.1.318 Type II Topp-Leone Inverted Kumaraswamy (TIITLIK) . . 1.1.319 New Power Topp-Leone Generated (NPTL-G) . . . . . . . 1.1.320 Modified Beta Gompertz (MBG) . . . . . . . . . . . . . . . 1.1.321 Exponentiated Weibull-Exponentiated Weibull (EW-EW) . . 1.1.322 Nadarajah Haghighi Topp Leaone-G (NHTL-G) . . . . . . . 1.1.323 Weibull Generalized Burr XII (WGBXII) . . . . . . . . . . 1.1.324 Extended Poisson Fréchet (EPFr) . . . . . . . . . . . . . . 1.1.325 Poisson Burr X -Fréchet (PBX-Fr) . . . . . . . . . . . . . . 1.1.326 Odd Log-Logistic Lindley-G (OLLLi-G) . . . . . . . . . . 1.1.327 Exponentiated Generalized Power Lindley (EG-PL) . . . . . 1.1.328 Generalized Odd Half-Cauchy-G (GOHC-G) . . . . . . . . 1.1.329 Generalized Kumaraswamy-G (GK-G) . . . . . . . . . . . 1.1.330 Alpha Power Inverted Exponential (APIE) . . . . . . . . . . 1.1.331 Alpha Power Inverse Weibull (APIW) . . . . . . . . . . . . 1.1.332 Kumarswamy Log-Logistic Weibull (KLLoGW) . . . . . . 1.1.333 Mrashall-Olkin Kappa (MOK) . . . . . . . . . . . . . . . . 1.1.334 Exponentiated Gumbel Exponential (EGuE) . . . . . . . . . 1.1.335 Type II Topp-Leone Power Lomax (TIITLPL) . . . . . . . . 1.1.336 XGamma-G (XG-G) . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

142 142 143 143 143 144 144 144 145 145 146 146 146 147 147 147 148 148 149 149 149 150 150 150 151 151 152 152 152 153 153 154 154 155 155 156 156 156 157 157 157 158 158 158 159

Contents 1.1.337 Weibull-Inverse Lomax (WIL) . . . . . . . . . . . . . . . . . . . 1.1.338 Kumaraswamy Alpha Power Inverted Exponential (KAPIE) . . . 1.1.339 Odd Generalized Exponential-Exponential (OGE-E) . . . . . . . 1.1.340 Topp-Leone-Lomax (TLLx) . . . . . . . . . . . . . . . . . . . . 1.1.341 Beta Generalized Exponentiated Fréchet (BGEF) . . . . . . . . . 1.1.342 Transmuted Generalized Inverted Exponential (TGIE) . . . . . . 1.1.343 Transmuted Generalized Power Weibull (TGPW) . . . . . . . . . 1.1.344 Size Biased Gamma Lindley (SBGaL) . . . . . . . . . . . . . . . 1.1.345 Skew t-Distribution of Three Degrees of Freedom (St-DTDF) . . 1.1.346 Logarithm Transformed Lomax (LTL) . . . . . . . . . . . . . . . 1.1.347 Exponentiated Generalized Power Function (EGPF) . . . . . . . 1.1.348 Gompertz Length Biased Exponential (Go-LBE) . . . . . . . . . 1.1.349 Gumbel-Burr XII (GUBXII) . . . . . . . . . . . . . . . . . . . . 1.1.350 Type II Power Topp-Leone Generated (TIIPTL-G) . . . . . . . . 1.1.351 Beta Kumaraswamy Marshall-Olkin-G (BKwMO-G) . . . . . . . 1.1.352 Beta Generalized Marshall-Olkin Kumarswamy-G (BGMOKw-G) 1.1.353 Beta Marshall-Olkin Kumarswamy-G (BMOKw-G) . . . . . . . 1.1.354 Zografos-Balakrishnan Burr XII (ZBBXII) . . . . . . . . . . . . 1.1.355 Beta Kumarswamy Burr Type X (BKBX) . . . . . . . . . . . . . 1.1.356 Beta Kumaraswamy Marshall-Olkin-G (BKwMO-G) . . . . . . . 1.1.357 Poisson Burr X Burr XII (PBXBXII) . . . . . . . . . . . . . . . 1.1.358 Transmuted Arcsine (TA) . . . . . . . . . . . . . . . . . . . . . 1.1.359 Poisson Burr X Generalized Lomax (PBXGL) . . . . . . . . . . 1.1.360 Transmuted Power Function (TPF) . . . . . . . . . . . . . . . . . 1.1.361 Kumaraswamy Moment Exponential (KwME) . . . . . . . . . . 1.1.362 Marshall-Olkin Length Biased Exponential (MOLBE) . . . . . . 1.1.363 Generalized Moment Exponential Power Series (GMEPS) . . . . 1.1.364 Truncated Exponential Skew Logistic (TESL) . . . . . . . . . . . 1.1.365 Balakrishnan Alpha Skew Normal2 (BASN2 ) . . . . . . . . . . . 1.1.366 Alpha Beta Skew Logistic-G (ABSLG) . . . . . . . . . . . . . . 1.1.367 Bimodal Alpha Skew LogisticG2 (BASLG2 ) . . . . . . . . . . . 1.1.368 Generalized Modified Exponential-G (GMEG) . . . . . . . . . . 1.1.369 Doubly Truncated Extreme Value Type I (DTEVTI) . . . . . . . 1.1.370 Lomax Exponential (LE) . . . . . . . . . . . . . . . . . . . . . . 1.1.371 Generalized Odd Log-Logistic Exponential (GOLLEx) . . . . . . 1.1.372 Extended Odd Weibull Exponential (EOWEx) . . . . . . . . . . . 1.1.373 New Libby-Novick (NLN) . . . . . . . . . . . . . . . . . . . . . 1.1.374 Modified Burr XII (MBXII) . . . . . . . . . . . . . . . . . . . . 1.1.375 Poisson Odd Generalized Exponential (POGE) . . . . . . . . . . 1.1.376 Modified Odd Weibull-G (MOW-G) . . . . . . . . . . . . . . . . 1.1.377 Generalized Uniform (GU) . . . . . . . . . . . . . . . . . . . . . 1.1.378 McDonald Modified Burr-III (McMB-III) . . . . . . . . . . . . . 1.1.379 Generalized Lindley (GL) . . . . . . . . . . . . . . . . . . . . . 1.1.380 Weibull Marshall-Olkin Lindley (WMOL) . . . . . . . . . . . . . 1.1.381 Generalized Inverse Marshall-Olkin (GIMO) . . . . . . . . . . .

xiii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159 159 160 160 161 161 161 162 162 162 163 163 163 164 164 165 165 166 166 167 167 167 168 168 169 169 170 170 170 171 171 171 172 172 173 173 174 174 174 175 175 175 176 176 177

xiv

Contents 1.1.382 Poisson Burr X Inverse Rayleigh (PBX-IR) . . . . . . . . . 1.1.383 Kumarswamy Reciprocal (KR) . . . . . . . . . . . . . . . . 1.1.384 Log-Weighted Pareto (LWP) . . . . . . . . . . . . . . . . . 1.1.385 Alpha Power Transformed Extended Exponential (APTEE) . 1.1.386 Lomax Exponentiated Weibull (LEW) . . . . . . . . . . . . 1.1.387 Intervened Geometric Compound (IGC) . . . . . . . . . . . 1.1.388 Intervened Negative Binomial Compound (INBC) . . . . . 1.1.389 Intervened Binomial Compound (IBC) . . . . . . . . . . . 1.1.390 Intervened Poisson Compound (IPC) . . . . . . . . . . . . 1.1.391 Odd Log-Logistic Exponential Gaussian (OLLExGa) . . . . 1.1.392 Gompertz Fréchet (GFr) . . . . . . . . . . . . . . . . . . . 1.1.393 Weighted Exponential Gompertz (WE-G) . . . . . . . . . . 1.1.394 Rayleigh Rayleigh (RR) . . . . . . . . . . . . . . . . . . . 1.1.395 Chen-G (CG) . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.396 Power Muth (PM) . . . . . . . . . . . . . . . . . . . . . . 1.1.397 Cubic Transmuted Power Function (CTPF) . . . . . . . . . 1.1.398 Weibull Alpha Power Inverted Exponential (WAPIE) . . . . 1.1.399 Exponentiated-Epsilon (E-Epsilon) . . . . . . . . . . . . . 1.1.400 Gompertz-Alpha Power Inverted Exponential (GAPIE) . . . 1.1.401 Gompertz Extended Generalized Exponential (G-EGE) . . . 1.1.402 Extended Odd Log-Logistic (EOLL-G) . . . . . . . . . . . 1.1.403 Generalized Odd Generalized Exponential G (GOGE-G) . . 1.1.404 Slash Maxwell (SM) . . . . . . . . . . . . . . . . . . . . . 1.1.405 Modified T-X (MT-X) . . . . . . . . . . . . . . . . . . . . 1.1.406 Double Truncated Transmuted Fréchet (DTTF) . . . . . . . 1.1.407 Burr-Hatke Logarithmic BurrXII (BH-BXII) . . . . . . . . 1.1.408 Odd Inverse Pareto-Exponential (OIPEx) . . . . . . . . . . 1.1.409 Poisson Rayleigh Generalized Lomax (PRGLx) . . . . . . . 1.1.410 Generalized Odd Log-Logistic-G (GOLL-G) . . . . . . . . 1.1.411 Marshall-Olkin Generalized Pareto (MOGP) . . . . . . . . 1.1.412 Log-Balakrishnan-Alpha-Skew-Normal (LBASN2 (α)) . . . 1.1.413 Flexible Additive Weibull (FAW) . . . . . . . . . . . . . . 1.1.414 Transmuted Weibull (TW) . . . . . . . . . . . . . . . . . . 1.1.415 Cubic Transmuted (CT) . . . . . . . . . . . . . . . . . . . 1.1.416 Transmuted Modified Weibull (TMW) . . . . . . . . . . . . 1.1.417 Transmuted Half Normal (THN) . . . . . . . . . . . . . . . 1.1.418 Cubic Rank Transmuted Fréchet (CRTF) . . . . . . . . . . 1.1.419 Weighted Garima (WG) . . . . . . . . . . . . . . . . . . . 1.1.420 Exponentiated Exponential Lomax (EEL) . . . . . . . . . . 1.1.421 Transmuted Half Normal (THN) . . . . . . . . . . . . . . . 1.1.422 Transmuted Ishita (TI) . . . . . . . . . . . . . . . . . . . . 1.1.423 Transmuted Generalized Extreme Value (TGEV) . . . . . . 1.1.424 Transmuted Burr Type X (TBX) . . . . . . . . . . . . . . . 1.1.425 Odd Lindley Exponentiated Weibull (OLi-EW) . . . . . . . 1.1.426 Transmuted Lomax-G (TL-G) . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 178 178 178 179 179 179 180 180 181 181 182 182 182 183 183 183 184 184 185 185 185 186 186 186 187 187 188 188 189 189 189 190 190 190 191 191 191 192 192 192 193 193 194 194

Contents 1.1.427 Power Lindley Geometric (PLG) . . . . . . . . . . . . . . . . . 1.1.428 Odd Lomax Exponential (OLxEx) . . . . . . . . . . . . . . . . 1.1.429 Jamal Weibull-X (JW-X) . . . . . . . . . . . . . . . . . . . . . 1.1.430 Topp-Leone Exponentiated-G (TLEx-G) . . . . . . . . . . . . . 1.1.431 New Extended Burr III (NEBIII) . . . . . . . . . . . . . . . . . 1.1.432 Zero Truncated Poisson Topp Leone Fréchet (ZTPTL-Fr) . . . . 1.1.433 Alpha Power Transformed Inverse Lindley (APTIL) . . . . . . 1.1.434 Beta Type I Generalized Half Logistic (BTIGHL) . . . . . . . . 1.1.435 Burr-Hatke Extended Burr XII (BHEBXII) . . . . . . . . . . . 1.1.436 Wrapped Lindley (WL) . . . . . . . . . . . . . . . . . . . . . . 1.1.437 Truncated Cauchy Power-G (TCP-G) . . . . . . . . . . . . . . 1.1.438 Slash Lindley-Weibull (SLW) . . . . . . . . . . . . . . . . . . 1.1.439 Risti´c-Balakrishnan Extended Exponential (RBEE) . . . . . . . 1.1.440 Exponentiated Generalized Standardized Gumbel (EGSGu) . . 1.1.441 Unit-Lindley (unit-Lindley) . . . . . . . . . . . . . . . . . . . 1.1.442 Odd Log-Logistic Dagum (OLLDa) . . . . . . . . . . . . . . . 1.1.443 Type I Half-Logistic Modified Weibull (TIHLMW) . . . . . . . 1.1.444 Odd Dagum (OddD-G) . . . . . . . . . . . . . . . . . . . . . . 1.1.445 Unit-Weibull (UW) . . . . . . . . . . . . . . . . . . . . . . . . 1.1.446 Generalized DUS Lindley (GDUSL) . . . . . . . . . . . . . . . 1.1.447 Poisson Rayleigh Burr XII (PRBXII) . . . . . . . . . . . . . . 1.1.448 Exponential-Gamma (EG) . . . . . . . . . . . . . . . . . . . . 1.1.449 Extended Beta Power Function (EBPF) . . . . . . . . . . . . . 1.1.450 Arcsine Exponentiated-X (ASE-X) . . . . . . . . . . . . . . . 1.1.451 Lindley Quasi XGamma (LQXG) . . . . . . . . . . . . . . . . 1.1.452 Type II Half Logistic Exponentiated Exponential (TIIHLEE) . . 1.1.453 Weighted Ishita (WI) . . . . . . . . . . . . . . . . . . . . . . . 1.1.454 Quasi Sujatha (QS) . . . . . . . . . . . . . . . . . . . . . . . . 1.1.455 Weighted Nakagami (WN) . . . . . . . . . . . . . . . . . . . . 1.1.456 Weighted Version of Generalized Inverse Weibull (WVGIW) . . 1.1.457 Weighted Inverse Lévy (WIL) . . . . . . . . . . . . . . . . . . 1.1.458 Exponentiated Length Biased Exponential (ELBE) . . . . . . . 1.1.459 Length-Biased Suja (LBS) . . . . . . . . . . . . . . . . . . . . 1.1.460 Alpha-Power Generalized Inverse Lindley (APGIL) . . . . . . . 1.1.461 Exponentiated Cubic Transmuted Exponential (ECTE) . . . . . 1.1.462 Transmuted Alpha Power Inverse Rayleigh (TAPIR) . . . . . . 1.1.463 New Beta Power Transformed (NBPT) . . . . . . . . . . . . . 1.1.464 Generalized Gamma Exponentiated Weibull (GGEW) . . . . . 1.1.465 Marshall-Olkin Exponential Gompertz (MOEGo) . . . . . . . . 1.1.466 Exponentiated Truncated Inverse Weibull-Generated (ETIW-G) 1.1.467 Lindley Rayleigh (LR) . . . . . . . . . . . . . . . . . . . . . . 1.1.468 Generalized Weibull Uniform (GWU) . . . . . . . . . . . . . . 1.1.469 Burr XII Uniform (BXIIU) . . . . . . . . . . . . . . . . . . . . 1.1.470 G-Fixed-Topp-Leone (G-FTL) . . . . . . . . . . . . . . . . . . 1.1.471 Generalized New Extended Weibull (GNEW) . . . . . . . . . .

xv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

194 195 195 195 196 196 196 197 197 198 198 198 199 199 199 200 200 201 201 201 202 202 202 203 203 204 204 204 205 205 205 205 206 206 207 207 207 208 208 209 209 209 210 210 210

xvi

Contents 1.1.472 New Exponential Trigonometric (NET) . . . . . . . . . . . . . . . 1.1.473 Power Log-Dagum (PLD) . . . . . . . . . . . . . . . . . . . . . . 1.1.474 Poly-Exponential Transformation (PET) . . . . . . . . . . . . . . . 1.1.475 New Weighted Transmuted Exponential (NWTE) . . . . . . . . . . 1.1.476 Kumarswamy Poisson-G (KwP-G) . . . . . . . . . . . . . . . . . . 1.1.477 Truncated Burr-G (TB-G) . . . . . . . . . . . . . . . . . . . . . . 1.1.478 Alpha Power Transformation Lomax (APTL) . . . . . . . . . . . . 1.1.479 Weighted Inverted Weibull (WIW) . . . . . . . . . . . . . . . . . . 1.1.480 Odd Exponential-Pareto IV (OEPIV) . . . . . . . . . . . . . . . . 1.1.481 T-Dagum{Y } (TD{Y }) . . . . . . . . . . . . . . . . . . . . . . . . 1.1.482 Generalized Odd Inverted Exponential-G (GOIE-G) . . . . . . . . 1.1.483 Marshall-Olkin Burr Exponential-2 (MOBE-2) . . . . . . . . . . . 1.1.484 Exponentiated Power Generalized Weibull Power Series (EPGWPS) 1.1.485 Exponentiated Transmuted Weibull Geometric (ETWG) . . . . . . 1.1.486 Complementary Exponential Geometric (CEG) . . . . . . . . . . . 1.1.487 Complementary Exponentiated Lomax-Poisson (CELP) . . . . . . 1.1.488 Alpha Power Transformed Log-Logistic (APTLL) . . . . . . . . . 1.1.489 T-Kumarswamy (T-K) . . . . . . . . . . . . . . . . . . . . . . . . 1.1.490 Zografos Balakrishnan Power Lindley (ZB-PL) . . . . . . . . . . . 1.1.491 Poisson Exponential-G (PE-G) . . . . . . . . . . . . . . . . . . . . 1.1.492 Perturbed Half-Normal (PHN) . . . . . . . . . . . . . . . . . . . . 1.1.493 Normal-Poisson (NP) . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.494 Extended Exponentiated Weibull (EEW) . . . . . . . . . . . . . . 1.1.495 New Family of Heavy Tailed (NFHT) . . . . . . . . . . . . . . . . 1.1.496 New Heavy Tailed Family of Claim (NHTFC) . . . . . . . . . . . . 1.1.497 New Beta Power Transformed (NBPT) . . . . . . . . . . . . . . . 1.1.498 Transmuted Type II Generalized Logistic (TTIIGL) . . . . . . . . . 1.1.499 Beta-Complementary Exponential Power Series (BCEPS) . . . . . 1.1.500 Zubair-G (Zubair-G) . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.501 Generalized Class (GC) . . . . . . . . . . . . . . . . . . . . . . . 1.1.502 Odd Lindley Half Logistics (OLiHL) . . . . . . . . . . . . . . . . 1.1.503 Modified Beta Generalized Linear Failure Rate (MBGLFR) . . . . 1.1.504 Truncated Weibull Fréchet (TWFr) . . . . . . . . . . . . . . . . . 1.1.505 Beta Poisson-G (BP-G) . . . . . . . . . . . . . . . . . . . . . . . . 1.1.506 Topp-Leone Marshall-Olkin-G (TLMO-G) . . . . . . . . . . . . . 1.1.507 Beta Modified Weibull Power Series (BMWPS) . . . . . . . . . . . 1.1.508 Fréchet Weibull (FW) . . . . . . . . . . . . . . . . . . . . . . . . 1.1.509 Odd Lomax Fréchet (OLxF) . . . . . . . . . . . . . . . . . . . . . 1.1.510 Burr X Weibull (BXW) . . . . . . . . . . . . . . . . . . . . . . . . 1.1.511 Beta Exponential Pareto (BEP) . . . . . . . . . . . . . . . . . . . . 1.1.512 Gompertz Flexible Weibull (GoFW) . . . . . . . . . . . . . . . . . 1.1.513 Type I Half-Logistic Rayleigh (TIHLR) . . . . . . . . . . . . . . . 1.1.514 Log-Beta Modified Weibull (LBMW) . . . . . . . . . . . . . . . . 1.1.515 Transmuted Exponential-G (TE-G) . . . . . . . . . . . . . . . . . 1.1.516 New Mixture of Exponential-Gamma (NMEG) . . . . . . . . . . .

211 211 211 212 212 213 213 213 214 214 214 215 215 216 216 217 217 217 218 218 219 219 219 220 220 220 221 221 221 222 222 223 223 223 224 224 225 225 225 226 226 227 227 227 228

Contents 1.1.517 New Generalized Akash (NGA) . . . . . . . . . . . . . . 1.1.518 T-R { Y } Power Series (T-R { Y } PS) . . . . . . . . . . 1.1.519 New Odd Log-Logistic Chen (NOLL-Ch) . . . . . . . . . 1.1.520 Transmuted Generalized Lindley (TGL) . . . . . . . . . . 1.1.521 Beta Exponentiated Weibull Geometric (BEWG) . . . . . 1.1.522 Beta Exponentiated Nadarajah-Haghighi (BENH) . . . . . 1.1.523 Exponentiated Additive Weibull (EAW) . . . . . . . . . . 1.1.524 Transmuted Kumaraswamy Lindley (TKL) . . . . . . . . 1.1.525 Weighted Power Lindley (WPL) . . . . . . . . . . . . . . 1.1.526 Marshall-Olkin Extended Power Lomax (MOEPL) . . . . 1.1.527 Unit Modified Burr-III (UMBIII) . . . . . . . . . . . . . 1.1.528 Reflected Power Function (RPF) . . . . . . . . . . . . . . 1.1.529 Transmuted Odd Fréchet-G (TOFr-G) . . . . . . . . . . . 1.1.530 Unit- Birnbaum-Saunders (UBS) . . . . . . . . . . . . . . 1.1.531 erf-G (erf-G) . . . . . . . . . . . . . . . . . . . . . . . . 1.1.532 New Unit-Lindley (NUL) . . . . . . . . . . . . . . . . . 1.1.533 Odd Generalized Exponentiated Inverse Lomax (OGE-IL) 1.1.534 Marshall-Olkin Inverse-Lomax (MO-IL) . . . . . . . . . 1.1.535 Odd Lindley-Rayleigh (OLR) . . . . . . . . . . . . . . . 1.1.536 Cubic Transmuted Gompertz (CTG) . . . . . . . . . . . . 1.1.537 Type II Topp-Leone-Power Lomax (TIITL-PL) . . . . . . 1.1.538 Odd Lindley Inverse Exponential (OLINEX) . . . . . . . 1.1.539 New Generalized Rayleigh (NGR) . . . . . . . . . . . . . 1.1.540 Marshall-Olkin Power Generalized Weibull (MOPGW) . . 1.1.541 Type II Topp-Leone Power Ishita (TIITLPI-G) . . . . . . 1.1.542 Weighted Inverse Nakagami (WINK) . . . . . . . . . . . 1.1.543 Generalized Marshall-Olkin Poisson-G (GMOP-G) . . . . 1.1.544 Half-Logistic XGamma (HLXG) . . . . . . . . . . . . . . 1.1.545 Log-Gamma-Generated (LGG1 ) . . . . . . . . . . . . . . 1.1.546 Log-Gamma-Generated (LGG2 ) . . . . . . . . . . . . . . 1.1.547 McDonald Gumbel (MG) . . . . . . . . . . . . . . . . . 1.1.548 Alpha-Beta Skew Logistic G (ABSLG) . . . . . . . . . . 1.1.549 Generalized Transmuted Poisson-G (GTPG) . . . . . . . 1.1.550 Generalized Marshall-Olkin Transmuted-G (GMOT-G) . . 1.1.551 Generalized Odd Linear Exponential (GOLE) . . . . . . . 1.1.552 Exponentiated Odd Chen-G (EOCh-G) . . . . . . . . . . 1.1.553 Transmuted Complementary Exponential Power (TCEP) . 1.1.554 Hyperbolic Cosine Weibull (HCW) . . . . . . . . . . . . 1.1.555 Burr X Exponential-G (BXE-G) . . . . . . . . . . . . . . 1.1.556 Extended Generalized Inverse Exponential (EGIEx) . . . 1.1.557 Weibull-Negative Binomial (WNB) . . . . . . . . . . . . 1.1.558 Unit Generalized Half Normal (UGHN) . . . . . . . . . . 1.1.559 Generalization of Exponential and Lindley (GEL) . . . . . 1.1.560 Unit Nadarajah-Haghighi Generated (UNH-G) . . . . . . 1.1.561 Lomax D function Generalized Weibull (LDGW) . . . . .

xvii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

228 228 229 229 230 230 230 231 231 231 232 232 232 233 233 234 234 234 235 235 236 236 236 237 237 238 238 238 239 239 239 240 240 240 241 241 242 242 243 243 243 244 244 244 245

xviii

Contents 1.1.562 Right Truncated Power Lomax (RTPL) . . . . . . . . . . . . . . 1.1.563 Exponentiated Garima (EG) . . . . . . . . . . . . . . . . . . . . 1.1.564 New Extended Weibull (NEW) . . . . . . . . . . . . . . . . . . . 1.1.565 Kumarswamy Sushila (KwS) . . . . . . . . . . . . . . . . . . . . 1.1.566 Alpha Power Transformed Inverse Lomax (APTIL) . . . . . . . . 1.1.567 New Extended-F (NE-F) . . . . . . . . . . . . . . . . . . . . . . 1.1.568 Topp-Leone Rayleigh (TLR) . . . . . . . . . . . . . . . . . . . . 1.1.569 Marshall-Olkin Topp Leone-G (MOTL-G) . . . . . . . . . . . . 1.1.570 Right Truncated-X (RT-X) . . . . . . . . . . . . . . . . . . . . . 1.1.571 Generalized Marshall-Olkin Inverse Lindley (GMOIL) . . . . . . 1.1.572 Generalized Reciprocal Exponential (GRE) . . . . . . . . . . . . 1.1.573 Generalized Transmuted Moment Exponential (GTME) . . . . . 1.1.574 Generalized Weighted Exponential (GWEx) . . . . . . . . . . . . 1.1.575 Exponentiated Transmuted Length-Biased Exponential (ETLBE) 1.1.576 Gamma Inverse Weibull (GIW) . . . . . . . . . . . . . . . . . . 1.1.577 Transmuted Alpha Power-G (TAPO-G) . . . . . . . . . . . . . . 1.1.578 Exponentiated Poisson-Exponential (EPE) . . . . . . . . . . . . . 1.1.579 Exponentiated Power Function (EPF) . . . . . . . . . . . . . . . 1.1.580 Alpha Logarithmic Transformed Weibull (ALTW) . . . . . . . . 1.1.581 Odd Exponentiated Half-Logistic Exponential (OEHLEx) . . . . 1.1.582 Odd Burr III Exponential (OBIIIE) . . . . . . . . . . . . . . . . 1.1.583 Rayleigh-Geometric (RG) . . . . . . . . . . . . . . . . . . . . . 1.1.584 Weibull Exponentiated Exponential (WEE) . . . . . . . . . . . . 1.1.585 Logistic Exponential (LE) . . . . . . . . . . . . . . . . . . . . . 1.1.586 Generalized Gamma-G (GG-G) . . . . . . . . . . . . . . . . . . 1.1.587 Exponentiated Shanker (E-Sh) . . . . . . . . . . . . . . . . . . . 1.1.588 Extended Generalized Lindley (EGL) . . . . . . . . . . . . . . . 1.1.589 Ristic-Balakrishnan Odd Log-Logistic-G (RBOLL-G) . . . . . . 1.1.590 Burr XII Exponentiated Exponential (BrXIIEE) . . . . . . . . . . 1.1.591 Mixture Pareto Log-Gamma (MPLG) . . . . . . . . . . . . . . . 1.1.592 Exponentiated Two Parameter Pranav (ETPP) . . . . . . . . . . . 1.1.593 New Lifetime Exponential-Weibull (NLTE-W) . . . . . . . . . . 1.1.594 New Cubic Rank Transmutation (NCRT) . . . . . . . . . . . . . 1.1.595 Alpha Power Exponentiated Exponential (APExE) . . . . . . . . 1.1.596 Odd Fréchet Inverse Lomax (OFIL) . . . . . . . . . . . . . . . . 1.1.597 Topp-Leone Power Lindley (TLPL) . . . . . . . . . . . . . . . . 1.1.598 Gull Alpha Power (GAP) . . . . . . . . . . . . . . . . . . . . . . 1.1.599 Modi Generator (MG) . . . . . . . . . . . . . . . . . . . . . . . 1.1.600 Cubic Transmuted Exponentiated Pareto-1 (CTEP-1-G) . . . . . 1.1.601 Cubic Transmuted Exponentiated Pareto-1 (CTEP-1-R) . . . . . . 1.1.602 Slashed Quasi-Gamma (SQG) . . . . . . . . . . . . . . . . . . . 1.1.603 Log-Epsilon-Skew Normal (LESN) . . . . . . . . . . . . . . . . 1.1.604 Generalized Odd Log-Logistic Log-Normal (GOLLLN) . . . . . 1.1.605 Slash Power Maxwell (SPM) . . . . . . . . . . . . . . . . . . . . 1.1.606 Modified Slashed Half-Normal (MSHN) . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245 246 246 246 247 247 247 248 248 248 249 249 250 250 251 251 252 252 252 253 253 254 254 254 255 255 256 256 257 257 258 258 258 259 259 259 260 260 260 261 261 261 262 262 262

Contents 1.1.607 One Parameter Polynomial Exponential-G (OPPE-G) . . . . . 1.1.608 Beta Burr Type X (BBX) . . . . . . . . . . . . . . . . . . . . 1.1.609 Gamma Burr Type X (GBX) . . . . . . . . . . . . . . . . . . 1.1.610 Weibull Burr Type X (WBX) . . . . . . . . . . . . . . . . . . 1.1.611 Extended Odd Weibull-G (ExOW-G) . . . . . . . . . . . . . 1.1.612 Sinh Inverted Exponential (SIE) . . . . . . . . . . . . . . . . 1.1.613 Exponential Skew-Normal (ESN) . . . . . . . . . . . . . . . 1.1.614 T-Burr (T-Burr{Y }) . . . . . . . . . . . . . . . . . . . . . . . 1.1.615 Transmuted General (T-G) . . . . . . . . . . . . . . . . . . . 1.1.616 Odd Gamma Weibull-Geometric (OGWG) . . . . . . . . . . 1.1.617 Generalized Inverted Kumarswamy Generated (GIKw-G) . . 1.1.618 Gamma Kumarswamy-G (GKw-G) . . . . . . . . . . . . . . 1.1.619 Weibull Burr XII (WBXII) . . . . . . . . . . . . . . . . . . . 1.1.620 Odd Generalized Gamma-G (OGG-G or GG-G) . . . . . . . . 1.1.621 Marshall-Olkin Odd Burr III-G (MOOB-G) . . . . . . . . . . 1.1.622 Gamma Power Half-Logistic (GPHL) . . . . . . . . . . . . . 1.1.623 Topp-Leone Weibull-Lomax (TLWLx) . . . . . . . . . . . . 1.1.624 Minimum Weibull-Burr (minWB) . . . . . . . . . . . . . . . 1.1.625 Box-Cox Gamma-G (BCG-G) . . . . . . . . . . . . . . . . . 1.1.626 Modified Beta Generalized Linear Failure Rate (MBGLFR) . 1.1.627 New Modified Burr III (NMBIII) . . . . . . . . . . . . . . . 1.1.628 Inverted Modified Lindley (IML) . . . . . . . . . . . . . . . 1.1.629 Type II General Inverse Exponential (TIIGIE) . . . . . . . . . 1.1.630 Exponentiated Half-Logistic Lomax (EHLLx) . . . . . . . . . 1.1.631 Generalized Gamma-Generalized Inverse Weibull (GG-GIW) 1.1.632 Log-Weighted Exponential (log-WE) . . . . . . . . . . . . . 1.1.633 Generalized Raised Cosine (GENRC) . . . . . . . . . . . . . 1.1.634 Sine Kumarswamy-G (SK-G) . . . . . . . . . . . . . . . . . 1.1.635 Extended Exp-G (EE-G) . . . . . . . . . . . . . . . . . . . . 1.1.636 Weighted Exponential (WE) . . . . . . . . . . . . . . . . . . 1.1.637 Lindley Negative-Binomial (LNB) . . . . . . . . . . . . . . . 1.1.638 Marshall-Olkin Transmuted-G (MOT-G) . . . . . . . . . . . 1.1.639 Ratio of Two Independent Weibull and Lindley (RTIWL) . . . 1.1.640 Product of Two Independent Weibull and Lindley (PTIWL) . 1.1.641 Modified Kies Generalized (MKi-G) . . . . . . . . . . . . . . 1.1.642 Ratio Exponentiated General (RE-G) . . . . . . . . . . . . . 1.1.643 Gompertz Exponential (GoEp) . . . . . . . . . . . . . . . . . 1.1.644 Minimum Gumbel Burr (minGuBu) . . . . . . . . . . . . . . 1.1.645 Transmuted Power Gompertz (TPG) . . . . . . . . . . . . . . 1.1.646 Burr X-G (BX-G) . . . . . . . . . . . . . . . . . . . . . . . . 1.1.647 Hamza (Hamza) . . . . . . . . . . . . . . . . . . . . . . . . 1.1.648 Inverse Lomax-G (IL-G) . . . . . . . . . . . . . . . . . . . . 1.1.649 Zubair-Inverse Lomax (ZIL) . . . . . . . . . . . . . . . . . . 1.1.650 Skew Scale Mixtures Normal (SSMN) . . . . . . . . . . . . . 1.1.651 Akash (Akash) . . . . . . . . . . . . . . . . . . . . . . . . .

xix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

263 263 264 264 265 265 266 266 266 267 267 268 268 269 269 270 270 271 271 271 272 272 273 273 274 274 274 275 275 276 276 276 277 277 277 278 278 279 279 279 280 280 280 281 281

Contents

xx

1.1.652 A Generalization of Sujatha (AGS) . . . . . . . . . . . . . . 1.1.653 Two-Parameter Sujatha (TPS) . . . . . . . . . . . . . . . . . 1.1.654 New Two-Parameter Sujatha (NTPS) . . . . . . . . . . . . . 1.1.655 Another Two-Parameter Sujatha (ATPS) . . . . . . . . . . . . 1.1.656 Lomax Inverse Weibull (LxIW) . . . . . . . . . . . . . . . . 1.1.657 Odd Burr Generalized Rayleigh (OBGR) . . . . . . . . . . . 1.1.658 Marshall-Olkin Lehmann Burr X (MOLBX) . . . . . . . . . 1.1.659 Transmuted Topp-Leone Weibull (TTL-W) . . . . . . . . . . 1.1.660 Burr X Generalized Burr XII (BXGBXII) . . . . . . . . . . . 1.1.661 Generalized Odd Generalized Exponential Fréchet (GOGEFr) 1.1.662 Topp-Leone Lindley (TLLi) . . . . . . . . . . . . . . . . . . 1.1.663 Logarithmic Transformed Inverse Weibull (LTIW) . . . . . . 1.1.664 Kumarswamy Alpha Power-G (KAP-G) . . . . . . . . . . . . 1.1.665 Flexible Weibull Burr XII (FWBXII) . . . . . . . . . . . . . 1.1.666 Extended Poisson Lomax (EPLx) . . . . . . . . . . . . . . . 1.1.667 Unit Johnson SU (UJSU ) . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

281 282 282 282 283 283 284 284 285 285 286 286 286 287 287 288

2 Characterizations of Distributions 2.1 Characterizations Based on Two Truncated Moments . . . . . . . . . . . . 2.2 Characterization in Terms of Hazard Function . . . . . . . . . . . . . . . . 2.3 Characterization in Terms of the Reverse (or Reversed) Hazard Function . . 2.4 Characterization Based on the Conditional Expectation of Certain Function of the Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . .

289 289 385 397

Appendix

415

References

417

About the Author

463

Index

465

407

Dedicated to my wife, Azam Niroomand-Rad and in memory of my beloved parents

Preface In designing a stochastic model for a particular modeling problem, an investigator will be vitally interested to know if their model fits the requirements of a specific underlying probability distribution. To this end, the investigator will rely on the characterizations of the selected distribution. Thus, the problem of characterizing a distribution is an important problem in various fields and has recently attracted the attention of many researchers. Consequently, various characterization results have been reported in the literature. These characterizations have been established in different directions. We present characterizations of 667 recently proposed distributions. This work is a continuation of our previous work, (Hamedani and Maadooliat, 2017) and (Hamedani, 2019) on the characterizations and infinite divisibility of recently introduced distributions. The current monograph may also serve as a source of preventing the reinvention and/or duplication of the existing distributions in the future. As pointed out in our papers cited here, a good number of proposed distributions have already been introduced in the literature. We believe the authors should do a detailed literature search before spending time on the already existing distributions. G. G. Hamedani Milwaukee, WI, US 2020

Chapter 1

Introduction In designing a stochastic model for a particular modeling problem, an investigator will be vitally interested to know if their model fits the requirements of a specific underlying probability distribution. To this end, the investigator will rely on the characterizations of the selected distribution. Generally speaking, the problem of characterizing a distribution is an important problem in various fields and has recently attracted the attention of many researchers. Consequently, various characterization results have been reported in the literature. These characterizations have been established in many different directions. The present work deals with the characterizations of a large number of new univariate continuous distributions as they became available to the author rather than the order of their importance. This work is a continuation of our previous works (Hamedani and Safavimanesh, 2017), (Hamedani, 2017), (Hamedani andMaadooliat, 2017), (Hamedani, 2018a), (Hamedani, 2018b) and (Hamedani 2019) on the characterizations and infinite divisibility of distributions introduced in 2016-2018. The current work and our previous published papers mentioned above may serve as a source of preventing the reinvention and/or duplication of the existing distributions in the future. As pointed out in our papers, a good number of proposed distributions have already been introduced in the literature. We believe the authors should do a detailed literature search before spending time on the already existing distributions. As mentioned in our previous works, in designing a stochastic model for a particular modeling problem, an investigator will be vitally interested to know if their model fits the requirements of a specific underlying probability distribution. To this end, the investigator will rely on the characterizations of the selected distribution. Thus, the problem of characterizing a distribution is an important problem in various fields and has recently attracted the attention of many researchers. Consequently, various characterization results have been reported in the literature. These characterizations have been established in different directions. This work deals with various characterizations of: 1) Extended Weighted Exponential (EWE) distribution of Mahdavi and Jabbari (2017); 2) Exponentiated Generalized Extended Pareto (EGEP) distribution of de Andrade and Zea (2018); 3) Inverse Weibull Generator (IWG) of distributions of Hassan and Nassr (2018); 4) Generalized Burr X-G (GBX-G) family of distributions of Aldahlan et al. (2018); 5) Generalized Lindley Power Series (GLPS) distribution of Rashid et al. (2018); 6) Transmuted Exponentiated Additive Weibull (TEAW) distribution of Nofal et al. (2018); 7) Exponentiated Weighted Expo-

2

G. G. Hamedani

nential (EWE) distribution of Oguntunde (2015); 8) New Weighted Exponential (NWE) distribution of Oguntunde et al. (2016); 9) Gompertz Lomax (GoLom) distribution of Oguntunde et al. (2017); 10) Chen’s Two-Parameter Exponential Power Life-Testing (CTPEPLT) distribution of Shakil et al. (2018); 11) Inverted Weighted Exponential (IWE) distribution of Oguntunde et al. (2018); 12) Kumaraswamy Marshall-Oklin Exponential (KMOE) distribution of George and Thobias (2018); 13) Kumaraswamy Weibull (KumW) of Cordeiro et al. (2010); 14) Kumaraswamy Half-Logistics (KH-L) distribution of Usman et al. (2017); 15) Transmuted Generalized Linear Exponential (TGLE) distribution of Elbatal et al. (2013); 16) a) Kumaraswamy-Chen (Kw-Chen); b) Kumaraswamy-XTG (KwXTG) and c) Kumaraswamy Flexible Weibull (Kw-FW) distributions of Nadarajah et al. (2014); 17) Exponentiated Generalized Inverted Exponential (EGIE) distribution of Fatima and Ahmad (2018); 18) Exponentiated Generalized Exponentiated Exponential (EGEE) distribution of Bukoye and Oyeyemi (2018); 19) Transmuted Exponentiated U-quadratic (TEUq) distribution of Muhammad and Suleiman (2019); 20) Generalized Gamma Burr III (GGBIII) distribution of Kehinde et al. (2018); 21) Beta Skew-t (BST) distribution of Basalamah et al. (2018); 22) A Class of Lindley and Weibull (ACLW) distributions of Alkarni (2016); 23) Generalized Inverted Kumaraswamy (GIKum) distribution of Iqbal et al. (2017); 24) Exponentiated Generalized Extended Pareto (EGEP) distribution of De Andrade and Zea (2018); 25) Benktander Type II (BType II) distribution of Kilany and Hassanein (2018); 26) Generalized Transmuted Fréchet (GTFr) distribution of Nofal and Ahsanullah (2018); 27) Beta Linear Failure Rate Power Series (BLFRPS) distribution of Makubate et al. (2018); 28) Odd Lomax-G (OLxG) distribution of Cordeiro et al. (2019); 29) Exponentiated Power Generalized Weibull (EPGW) distribution of Péna-Ramírez et al. (2018); 30) Lindley Weibull (LiW) distribution of Cordeiro et al. (2018); 31) MarshallOlkin Alpha Power (MOAP) distribution of Nassar et al. (2018); 32) Zero Spiked Gamma Weibull (ZSGW) distribution of Hashimoto et al. (2018); 33) Inverted Nadarajah-Haghighi (INH) distribution of Tahir et al. (2018); 34) Marshall-Olkin Generated Gamma (MOGG) distribution of Barriga et al. (2018); 35) Gamma Generalized Normal (GGN) distribution of Cordeiro et al. (2019); 36) Transmuted Transmuted-G (TTG) family of distributions of Mansour et al. (2019); 37) Power Binomial Exponential 2 (PBE2) distribution of Habibi and Asgharzadeh (2018); 38) Beta Burr III (BBIII) distribution of Gomes et al. (2013); 39) Muth Generated (MG) family of distributions of Almarashi and Elgarhy (2018); 40) Weibull-Lindley (WLn) distribution of Ieren et al. (2018); 41) Weibull-G Power Series (WGPS) of Mashabe et al. (2019); 42) Three Parameter Generalized Lindley (TPGL) distribution of Ekhosuehi and Opone (2018); 43) Odd Lindley Exponentiated Weibull (OLEW) distribution of Aboraya (2018); 44) Extended Odd Fréchet-G (EOF-G) distribution of Nasuri (2018); 45) Alpha Power Transformation Poisson Lindley (APTPL) distribution of Hassan et al. (2019); 46) Alpha Logarithm Transmuted Fréchet (ALTF) distribution of Dey et al. (2019); 47) Burr-Weibull Power Series (BWPS) distribution of Oluyede et al. (2019); 48) Zografos-Balakrishnan Fréchet (ZBFr) distribution of Charkraborty et al. (2019); 49) Cubic Transmuted Weibull (CTW) distribution of Abd AL-Kadim and Mohammed (2017); 50) Cubic Rank Transmuted Kumaraswamy (CRTKw) distribution of Saraço˘glu and Tani¸s (2018); 51) Cubic Transmuted Weibull (CTW) distribution of Rahman et al. (2019); 52) Type II Kumaraswamy Half Logistic-Generated (TIIKwHL-G) distribution of El-Sherpieny and Elsehetry (2019); 53) Odd Log-Logistic Generalized Inverse Gaussian (OLLGIG) dis-

Introduction

3

tribution of Vasconcelos et al. (2019); 54) General Class (GC) of distributions of Cortés et al. (2018); 55) Poisson Burr Type X Log-Logistic (PBXLL) distribution of Almamy (2019); 56) New Odd Generalized Exponential-Exponential (NOGE-E) distribution of Badamasi and Singh (2018); 57) Kumaraswamy Extension Exponential (KEE) distribution of Elbatal et al. (2018); 58) Extended Enlarg Transmuted Exponential (EETE) distribution of Okorie et al. (2017); 59) Marshall-Olkin Extended Power Function (MOEPF) distribution of Okorie et al. (2017); 60) Generalized Odd Log-Logistic Exponential (GOLLE) distribution of Qoshja and Muça (2018); 61) Exponentiated Transmuted Power Function (ETPF) distribution of Usman et al. (2018); 62) Type II Half Logistic Weibull (TIIHLW) distribution of Hassan et al. (2018); 63) Exponentiated Generalized Inverse Rayleigh (EGIR) distribution of Fatima et al. (2018); 64) Generalized Transmuted Gompertz-Makeham (GTGM) distribution of Riffi (2018); 65) Jamal Weibull-X (JW-X) family of distributions of Jamal and Nasir (2019); 66) Nasir Logistic-X (NL-X) family of distributions of Jamal and Nasir (2019); 67) Jamal Logistics-X (JL-X) family of distributions of Jamal and Nasir (2019); 68) Nasir Weibull-Generalized (NW-G) family of distributions of Jamal and Nasir (2019); 69) Weibull Pareto (WP) distribution of Tahir et al. (2015); 70) Slashed Power-Lindley (SPL) distribution of Iriarte and Rojas (2019); 71) Exponentiated Generalized Pareto (EGP) distribution of Lee and Kim (2019).; 72) Inverse Weighted Lindley (IWL) distribution of Ramos et al. (2019); 73) Unit-Inverse Gaussian (UIG) distribution of Ghitany et al. (2019); 74) Weibull-Moment Exponential (WME) distribution of Hashmi et al. (2016); 75) Generalized Odd Burr III-G (GOBIII-G) distribution of Ahsan ul Haq et al. (2018); 76) Generalized Odd Fréchet -G (GOFr-G) family of distributions of Ahsan ul Haq et al. (2018); 77) Type I Half Logistic Power Lindley (TIHLPL) distribution of Elbatal et al. (2018); 78) New AlphaPower Transformation (NAPT) of Elbatal et al. (2019); 79) Functional Weighted Exponential (FWE) of Bouali et al. (2019); 80) Odd Burr III G-Negative Binomial (OBIIIGNB) distribution of Jamal et al. (2019); 81) Type I Half-Logistic Exponential (TIHLE) distribution of Almarashi et al. (2019); 82) Unit-Marshall-Olkin Extended Exponential (UMOEE) distribution of Ghosh et al. (2019); 83) Generalization of Two-Parameter Lindley (GTPL) distribution of Shanker et al. (2019); 84) Odd Lindley Fréchet (OLiFr) distribution of Mansour et al. (2018); 85) Topp-Leone Mukherjee-Islam (TLMI) distribution of Al-Omari and Gharaibeh (2018); 86) Weibull-Lomax (WL) distribution of Hashmi and Gull (2018); 87) Zero Truncated Poisson Topp-Leone Exponentiated Weibull (ZTPTLEW) distribution of Refaie (2018); 88) Zero Truncated Poisson Topp-Leone Burr XII (ZTPTLBXII) distribution of Yousof et al. (2019); 89) General Transmuted Family (GTF) of distributions of Rahman et al. (2018); 90) Poisson Topp-Leone Inverse Weibull (PTLIW) distribution of Al-mualim (2019); 91) Odd Burr-G Poisson (OBGP) distribution of Nasir et al. (2018); 92) Exponentiated Mukherjee-Islam (EMI) distribution of Rather and Subramanian (2018); 93) Generalized Transmuted Power Function (GTPF) distribution of Abdul-Moniem and Diab (2018); 94) Poisson Exponentiated Erlang-Truncated Exponential (PEETE) distribution of Nasiru et al. (2018); 95) Minimum Guarantee Lindley (MGL) distribution of Kumar et al. (2018); 96) Inverted Beta (IB) and Inverted Beta Lindley (IBL) distributions of Kilany and Atallah (2018); 97) Cubic Transmuted Pareto (CTP) distribution of Ansari and Eledum (2018); 98) Zero Truncated Poisson Topp Leone Weibull (ZTPTLW) distribution of Thm (2018); 99) A New (AN) distribution of Doostmoradi (2018); 100) Exponentiated Kumarsawamy-G (EKw-G) class of distributions of Silva et al. (2019); 101) Log-Sinh Cauchy Promotion

4

G. G. Hamedani

(LSCp) time distribution of Ramires et al. (2017); 102) Modified Beta Modified-Weibull (MBMW) distribution of Saboor et al. (2018); 103) Kumaraswamy Generalized Linear Exponential (Kw-GLE) distribution of Yusuf and Qureshi (2019); 104) Weighted Inverse Gamma (WIG) distribution of Ahmad and Ahmad (2019); 105) Odd Log-Logistic Generalized Half-Normal Poisson (OLLGHNP) distribution of Lak et al. (2018); 106) Topp-Leone Weighted Weibull (TLWW) distribution of Abbas et al. (2019); 107) Burr-Hatke-G (BHG) family of distributions of Yousof et al. (2018); 108) Burr-Hatke Exponentiated Weibull (BHEW) distribution of Aboraya (2019); 109) Generalized Log-Lindley (GLL) distribution of Song and Wang (2019); 110) Weibull Generalized Log-Logistic (WGLL) distribution of Abouelmagd eta al. (2019); 111) Transmuted Generalized Odd Generalized Exponential-G (TGOGE-G) distribution of Reyad et al. (2019); 112) New Three Parameter Paralogistic (NTPL) distribution of Idemudia and Ekhosuehi (2019); 113) Janardan-Power Series (JPS) distribution of Shekari et al. (2019); 114) Exponentiated Generalized Extended Gompertz (EGEG) distribution of De Andrade et al. (2019); 115) Power-Exponential Hazard Rate (P-EHR) distribution of Tarvirdizade and Nematollahi (2019); 116) Inverse Power Lomax (IPL) distribution of Hassan and Abd-Allah (2019); 117) Exponentiated KumaraswamyWeibull (EK-W) distribution of Eissa, (2017); 118) Power Function Power Series (PFPS) class of distributions of Hassan and Assar (2019); 119) Exponentiated Negative Binomial (ENB) distribution of Hussain et al. (2019); 120) Burr-Hatke Exponential (BHE) distribution of Yadav et al. (2019); 121) X Gamma Weibull (XGW) distribution of Yousof et al. (2019); 122) Exponentiated Exponential Logistic (EEL) distribution of Ghosh and Alzaatreh (2018); 123) Reduced New Modified Weibull (RNMW) distribution of Almalki (2018); 124) Extended Weibull-G (EW-G) distribution of Korkmaz (2019); 125) Composite Generalizers of Weibull (CGW) distribution of Aryal et al. (2019); 126) Odd Log-Logistic Exponentiated Gumbel (OLLEGu) distribution of Alizadeh et al. (2018); 127) Harris Extended Lindley (HEL) of Cordeiro et al. (2019); 128) Odd Burr III Weibull (OBIIIW) distribution of Usman and Ahsan ul Haq (2019); 129) Power Lindley Generated (PLG) family of distributions of Hassan and Nassr (2019); 130) Inverse Gompertz (IG) distribution of Eliwa et al. (2019); 131) Hyperbolic Sine Rayleigh (HS-R) distribution of Ahmad (2019); 132) Type II Topp-Leone Generated (TIITL-G) family of distributions of Elgarhy et al. (2018); 133) Exponentiated Inverse Rayleigh (EIR) distribution of Rao and Mbwambo (2019); 134) Exponentiated New Weighted Weibull (ENWW) distribution of Elcherpieny et al. (2017); 135) Beta Transmuted Weighted Exponential (BTWE) distribution of Abdelall (2017); 136) Odd Lindley Exponentiated Weibull (OLi-EW) distribution of Refaie (2019); 137) Modified Weibull-G (MW-G) family of distributions of Abdelall (2019); 138) Compound Gamma and Lindley (GaL) distribution of Abdi et al. (2019); 139) Odd Weibull (OW) distribution of Mirzaei et a. (2019); 140) Topp-Leone Generalized Inverted Kumarswamy (TLGIKw) distribution of Reyad et al. (2019); 141) Marshall-Olkin Burr X (MOBX) family of distributions of Jamal et al. (2017); 142) New Extended Generalized Burr III (NEGBIII) family of distribution of Jamal et al. (2018); 143) Generalized Inverse Weibull-Generalized Inverse Weibull (GIW-GIW) distribution of Abid et al. (2019); 144) Exponentiated Burr XII Power Series (EBXIIPS) distribution of Nasir et al. (2019); 145) Burr XII Weibull Logarithmic (BWL) distribution of Oluyede et al. (2019); 146) Odd Log-Logistic Geometric Normal (OLLGN) distribution of Prataviera et al. (2019); 147) Marshall-Olkin Extended Flexible Weibull (MOEFW) distribution of Cordeiro et al.

Introduction

5

(2019); 148) Topp-Leone Inverse Weibull (TLIW) distribution of Abbas et al. (2017); 149) Odd Log-Logistic Marshall-Olkin Power Lindley (OLLMOPL) distribution of Alizadeh et al. (2018); 150) Topp-Leone Lomax (TLLo) distribution of Oguntunde et al. (2019); 151) Exponentiated Topp-Leone (ETL) distribution of Pourdarvish et al. (2015); 152) ToppLeone Generator (TLG) of distributions of Sangsanit and Bodhisuwan (2016); 153) ToppLeone Generated q-Exponential (TLG-qE) distribution of Sebastian et al. (2019); 154) Odd Hyperbolic Cosine KG (OHC-KG) distribution of Kharazmi et al.(2019); 155) Generalized Gudermannian (GG) distribution of Altun (2019); 156) New Extended Alpha Power Transformed (NEAPT) family of distributions of Ahmad et al. (2018); 157) Exponentiated Odd Log-Logistic-G (EOLL-G) family of distributions of Alizadeh et al. (2018); 158) Zero Truncated Poisson Topp Leone Weibull (ZTPTLW) distribution of Abouelmagd (2019); 159) Type II Generalized Topp-Leone-G (TIIGTL-G) distribution of Hassan et al. (2019); 160) Burr XII Exponentiated Weibull (BXIIEW) distribution of Hamed (2018); 161) Generalized Odd Log-Logistics Inverse Weibull (GOLLIW) distribution of Elbiely and Yousof (2019); 162) Centered Skew-Normal Birnbaum-Saunders (CSNBS) distribution of Chaves et al. (2019); 163) Logarithmic Kumarswamy (LKu) family of distributions of Ahmad (2019); 164) Marshall-Olkin Odd Lindley G (MOOL-G) family of distributions of Jamal et al. (2019); 165) Generalized Transmuted Fréchet (GTF) distribution of Riffi et al. (2019); 166) Weibull Generalized Exponentiated Weibull (WGEW) distribution of Hamed (2018); 167) Odd Log-Logistic Poisson-G (OLLP-G) family of distributions of Alizadeh et al. (2018); 168) New Lindley Exponential (NLE) distribution of Oguntunde et al. (2016); 169) Odd Log-Logistic Exponentiated Weibull (OLLEW) distribution of Afify et al. (2018); 170) Another Odd Log-Logistic Logarithmic (AOLLL-G) class of distributions of Alizadeh et al. (2018); 171) Type I Half-Logistic (TIHL-G) family of distributions of Cordeiro et al. (2015); 172) Odd Burr- Generalized (OBu-G) family of distributions of Alizadeh et al. (2017); 173) Odd Log-Logistic Topp-Leone G (OLLTL-G) family of distributions of Alizadeh et al. (2017); 174) Topp-Leone Odd Lindley-G (TLOL-G) family of distributions of Reyad et al. (2018); 175) Odd Log-Logistic Log-Normal (OLL-LN) distribution of Ozel et al. (2018); 176) Odd Log-Logistic Generalized Gompertz (OLLGG) distribution of Alizadeh et al. (2019); 177) New Odd Log-Logistic Half-Logistic (NOLLHL) distribution of Alizadeh et al. (2019); 178) Zografos-Balakrishnan Odd Log-Logistic Generalized Half-Normal (ZNOLL-GHN) distribution of Mozafari et al. (2019); 179) Odd Log-Logistic Marshall-Olkin Generalized Half-Normal (OLLMOGHN) distribution of Korkmaz et al. (2019); 180) Power-Linear Hazard Rate (P-LHR) distribution of Tarvirdizade and Nematollahi (2019); 181) Alpha-Power Pareto (APP) distribution of Ihtisham et al. (2019); 182) Exponentiated Odd Log-Logistic Normal (EOLLN) distribution of Altun et al. (2017); 183) Extended Odd Fréchet-G (EOF-G) family of distributions of Yousof et al. (2019); 184) Log-Odd Log-Logisticc Birnbaum-Saunders-Poisson (OLLBSP) distribution of Cordero et al. (2018); 185) Zografos-Balakrishnan Lindley-Poisson (ZB-LP) distribution of Oluyede et al. (2019); 186) One-Parameter Weibull-Type (1P-Weibull) distribution of Alexopoulos (2019); 187) Beta Odd Lindley-G (BOL-G) family of distributions of Chipepa et al. (2019); 188) A Distribution For Instantaneous Failures (ADFIF) of Ramos and Louzada (2019); 189) Generalized Inverse Weibull- Generalized Inverse Weibull (GIWGIW) distribution of Abid et al. (2019); 190) Beta Lindley Geometric (BLGc) distribution of Elbatal and Khalil (2019); 191) Odd Log-Logistic Generalized Half-Normal (OLLGHN)

6

G. G. Hamedani

distribution of Abdi et al. (2019); 192) Odd Lindley Lomax (OLLo) distribution of Ali et al. (2019); 193) Burr X Exponentiated Lomax (BrXELx) distribution of Aboraya (2019); 194) Extended Normal (EN) distribution of Lima et al. (2019); 195) Poisson-X (P-X) family of distributions of Tahir et al. (2016); 196) α-Power Transmuted Generalized Exponential (αPTGE) distribution of Dey et al. (2017); 197) Weibull Exponentiated Exponential (WEE) distribution of Zubair et al. (2018); 198) Weighted Modified Weibull (WMW) distribution of Khan et al. (2019); 199) Poisson Nadarajah-Haghighi (PNH) distribution of Mansoor et al. (2019); 200) Poisson Burr X Weibull (PBrXW) distribution of Abouelmagd et al. (2019); 201) Marshall-Olkin Extended Exponential (MOEE) distribution of Abu-Youssef et al. (2015); 202) New Generalization of Weibull-Exponential (NGWE) distribution of ZeinEldin and Elgarhy (2018); 203) Odd Fréchet Inverse Weibull (OFIW) distribution of Fayomi (2019); 204) Topp-Leone-G Poisson (TL-GP) class of distributions of Abouelmagd et al. (2019); 205) Alpha Power Transmuted Extended Exponential (APTEE) distribution of Hassan et al. (2019); 206) Odd Fréchet Inverse Rayleigh (OFIR) distribution of Elgarhy and Alrajhi (2019); 207) Odd Inverse Pareto-G (OIP-G) class of distributions of Aldahlan et al. (2019); 208) Type I Half-Logistic Burr X (TIHLBX ) distribution of Shrahili et al. (2019); 209) Geometric Lindley Poisson 1 (GLP1) distribution of Abd Elrazil and Mansour (2019); 210) Odd Fréchet Inverse Exponential (OFIE) distribution of Alrajhi (2019); 211) Truncated Weibull Power Lomax (TWPL) distribution of Al-Marzouki (2019); 212) Generalized Extended Inverse Weibull (GEIW) distribution of Hanagal and Bhalerao (2019); 213) Lomax-Lindley (L-L) distribution of Abdelall (2019); 214) Generalized GompertzGeneralized Gompertz (GG-GG) distribution of Boshi et al. (2019); 215) Exponentiated Weibull Weibull (EWW) distribution of Hassan and Elgarhy (2019); 216) WeibullInverse Lomax (WIL) distribution of Falgore et al. (2019); 217) Burr XII Inverse Rayleigh (BXII-IR) distribution of Goual and Yousof (2019); 218) Kumarswamy Type I Half Logistic (KwTIHL-G) family of distributions of El-Sherpieny and Elsehetry (2019); 219) Odd Generalized Exponential Power Function (OGEPF) distribution of Hassan et al. (2019); 220) Generalized Odd Lomax Generated (GOLG) family of distributions of Marzouk et al. (2019); 221) Inverse Weibull Geometric (IWG) distribution of Chakrabarty and Chowdhury (2019); 222) Inverse Weibull Poisson (IWP) distribution of Chakrabarty and Chowdhury (2019); 223) Transmuted Four Parameters Generalized Log-Logistic (TFPGLL) distribution of Adeyinka and Olapade (2019); 224) Burr X-Kumaraswamy (BXKw) distribution of Sanusi et al. (2019); 225) Transmuted Generalized Gamma (TGG) distribution of Saboor et al. (2019); 226) Generalized Extended Exponential-Weibull (GEEW) distribution of Shakhatreh et al. (2019); 227) Nadarajah-Haghigh Geometric (NHG) distribution of Marinho et al. (2019); 228) Nadarajah-Haghighi Lindley (NHL) distribution of Pena-Ramirez et al. (2019); 229) Exponentiated Log-Sinh Cauchy (ELSC) distribution of Ramires et al. (2018); 230) Modified Fréchet (MFr) distribution of Tablada and Cordeiro (2017); 231) Cubic Transmuted Uniform (CTU) distribution of Rahman et al. (2019); 232) Beta-G Poisson (BGP) distribution of Aryal et al. (2019); 233) Cosine-Sine Transformation (CST) distribution of Chesneau et al. (2019); 234) Generalized Burr XII Power Series (GBXIIPS) class of distributions of Elbatal et al. (2019); 235) Transmuted Extended Exponential (TEE) distribution of Kumar and Kumar (2019); 236) Exponentiated Generalized Power Series (EGPS) family of distributions of Nasiru et al. (2019); 237) Marshall-Olkin Alpha Power Inverse Exponential (MOAPIE) distribution of Basheer (2019); 238) Transmuted

Introduction

7

Topp Leone Exponentiated Fréchet (TTLEFr) distribution of Khalil (2019); 239) Generalized Gudermannian (GG) distribution of Altun (2019), simpler characterizations than 155); 240) Alpha Skew Generalized Gudermannian (ASGG) distribution of Altun (2019); 241) Truncated-Logistic Skew-Symmetric (TLSS) distribution of Ghosh and Ng (2019); 242) Combined Exponential-Normal {Generalized Weibull} (CE-N{GW}) distribution of Aljarrah et al. (2019); 243) Type II Half Logistic Ibrahim (TIIHLI) distribution of AbdulMoniem and Seham (2019); 244) Modified Generalized Marshall-Olkin (MGMO) family of distributions of Aslam et al. (2019); 245) Transmuted Extended Lomax (TEL) distribution of Momenkhan (2019); 246) Topp-Leone Odd Log-Logistic Exponential (TLOLLEx) distribution of Afify et al. (2019); 247) Odd Birnbaum-Saunders (OBS) distribution of Ortega et al. (2016); 248) Gamma Dual Weibull (Γ2 -W) distribution of Castellares and Lemonte (2016); 249) Quasi XGamma-Poisson (QXGP) distribution of Sen et al. (2018); 250) Modified Extended Generalized Exponential (MEGE) distribution of Okoli et al. (2016); 251) Exponentiated Nadarajah Haghighi Poisson (ENHP) distribution of Diouma Sira KA et al. (2019); 252) Extended New Generalized Exponential (ENGE of ENEGE as used by the authors) distribution of Zelibe et al. (2019); 253) Transmuted Singh-Maddala (TSM) distribution of Shahzad et al. (2017); 254) Transmuted Half Logistic (THL) distribution of Samuel and Kehinde (2019); 255) Kumarswamy Exponentiated U-Quadratic (KwEUQ) distribution of Muhammad et al. (2018); 256) New Generalized Transmuted Inverse Exponential (NGT-IE) distribution of Uwadi and Nwaezza (2018); 257) Transmuted Lomax Exponential (TLE) distribution of Kuje and Lasisi (2019); 258) Extended Pranav (EP) distribution of Uwaeme et al. (2018); 259) Generalized Odd Lindley-G (GOLi-G) family of distributions of Afify et al. (2019); 260) Leaned Normal (LN) distribution of Sen and Chandra (2016); 261) XGamma (XG) distribution of Sen et al. (2016); 262) Quasi XGammaGeometric (QXGGc) distribution of Sen et al. (2019); 263) Marshall-Olkin Modified Burr III (MOMBIII) distribution of Ahsan ul Haq et al. (2019); 264) Type II Exponentiated Half Logistic Generated (TIIEHL-G) family of distributions of Al-Mofleh et al. (2019); 265) Weighted XGamma (WXG) distribution of Sen et al. (2017); 266) Length Biased XGamma (LBXG) distribution of Sen et al. (2017); 267) Wrapped XGamma (WRXG) distribution of Al-Mofleh and Sen (2019); 268) Linearly Decreasing Stress Weibull (LDSWeibull) distribution of Barnard et al. (2019); 269) L-Logistic (L-Logistic) distribution of da Paz et al. (2019); 270) Normal Generalized Hyperbolic Secant (NGHS) distribution of Al-Mofleh (2019); 271) Generalized Inverse Pareto-G (GIP-G) distribution of Al-Mofleh and Afify (2019); 272) Kumarsawamy Marshall-Olkin Modified Weibull (KMOMW) distribution of Elbatal and Al-Mofleh (2019); 273) Wrapped Lindley-Exponential (WRLE) distribution of Al-Mofleh (2019); 274) Odoma distribution of Odom and Ijomah (2019); 275) Wibull Inverse Lomax (WIL) distribution of Hassan and Mohamed (2019); 276) Poisson Burr X Pareto Type II (PBXPTII) distribution of Abdelkhalek (2019); 277) Burr XII Fréchet (BrXIIFr) distribution of Ibrahim (2019); 278) Burr X Fréchet (BrXFr) distribution of Jahanshahi et al. (2019); 279) Odd Generalized Exponential Type-I Generalized Half Logistic (OGETIGHL) distribution of Lakshmi and Anjaneyulu (2018); 280) Burr X Nadarajah Haghighi (BXNH) distribution of Elsayed and Yousof (2019); 281) Topp-Leone Compound Rayleigh (TLCR) distribution of Rasheed (2019); 282) Dual Exponentiated Weibull (DEW) distribution of Elbiely (2019); 283) unit-Improved Second-Degree Lindley (unit-ISDL) distribution of Altun and Cordeiro (2019); 284) Kumaraswamy Odd Lindley-G (KOL-G) distribution

8

G. G. Hamedani

of Chipepa et al. (2019); 285) Transmuted Type I Generalized Logistic (TTIGL) distribution of Adeyinka (2019); 286) Truncated Discrete Linnik Weibull (TDLW) distribution of Jayakumar and Sankaran (2019); 287) Raised Cosine (RC) distribution; see King (2017); 288) Transmuted Exponentiated Weibull (TEW) distribution of Khan et al. (2019); 289) Generalized Marshall-Olkin Extended Burr-III (GMOBIII) of Chakraborty et al. (2019); 290) Modified Beta Linear Exponential (MBLE) distribution of Bakouch et al. (2019); 291) Zero-Truncated Poisson-Power Function (ZTPPF) distribution of Okorie et al. (2019); 292) Cubic Transmuted Pareto (CTP) distribution of Rahman et al. (2019); 293) Inverse XGamma (IXG) distribution of Yadav et al. (2019); 294) Alpha-Power Transformed Lindley (αPTL) distribution of Dey et al. (2019); 295) Log-Odd Logistic-Weibull (LOLW) distribution of Ortega et al. (2019); 296) Skewed Generalized Logistic (SGL) distribution of Theodossiou (2019); 297) Weighted T-X (WTX) family of distributions of Ahmad et al. (2019); 298) Zero-Truncated Poisson Exponentiated Gamma (ZTPEG) distribution of Aguilar et al. (2019); 299) Power Gompertz (PG) distribution of Ieren et al. (2019); 300) Transmuted New Weibull Pareto (TNWP) distribution of Al-Omari et al. (2019); 301) Lomax Weibull (LoW) distribution of Apam et al. (2019); 302) Odd Log-Logistic Geometric-G (OLLG-G) family of distributions of Lima et al. (2019); 303) MarshallOlkin Extended Quasi Lindley (MOEQL) distribution of Udoudo and Etuk (2018); 304) Poisson Rayleigh Log-Logistic (PRLL) distribution of Aboraya (2019); 305) Generalized Marshall-Olkin Extended Burr XII (GMOBXII) family of distributions of Handique and Chakraborty (2018); 306) Type I New Heavy Tailed Weibull (TINHT-W) distribution of Ahmad et al. (2019); 307) Arcsine Weibull (AS-W) distribution of Ahmad et al. (2019); 308) Truncated Inverted Kumaraswamy Generated (TIK-G) family of distributions of Bantan et al. (2019); 309) Lomax Gompertz-Makeham (LOGOMA) distribution of Eraikhuemen et al. (2019); 310) Exponential Transmuted Fréchet (ETF) distribution of Pillai and Moolath (2019); 311) Marshall-Olkin Extended Weibull Exponential (MOEWE) distribution of Chinazom et al. (2019); 312) Extended Log-Logistic (ELL) of Lima and Cordeiro (2017); 313) Log-Odd Normal Generalized (LONG) family of distributions of Zubair et al. (2019); 314) Marshall-Olkin Generalized Burr XII (MOGBXII) distribution of Afify and Abdellatif (2020); 315) Generalized Inverse Lindley (GIL) and Generalized Inverse Lindley Power Series (GILPS) distributions of Alkarni (2019); 316) Normal-C (N-C) class of distributions of Silveira et al. (2019); 317) Extended Power Lindley-G (EPL-G) family of distributions of Reyad et al. (2019); 318) Type II Topp-Leone Inverted Kumaraswamy (TIITLIK) distribution of ZeinEldin et al. (2019); 319) New Power Topp-Leone Generated (NPTL-G) family of distributions of Bantan et al. (2019); 320) Modified Beta Gompertz (MBG) distribution of Elbatal et al. (2019); 321) Exponentiated Weibull-Exponentiated Weibull (EW-EW) distribution of A–Noor et al. (2019); 322) Nadarajah Haghighi Topp Leaone-G (NHTL-G) family of distributions of Reyad et al. (2019); 323) Weibull Generalized Burr XII (WGBXII) distribution of Aborar and Butt (2019); 324) Extended Poisson Fréchet (EPFr) distribution of Khalil and Rezk (2019); 325) Poisson Burr X -Fréchet (PBX-Fr) distribution of Yousof et al. (2019); 326) Odd Log-Logistic Lindley-G (OLLLiG) family of distributions of Alizadeh et al. (2020); 327) Exponentiated Generalized Power Lindley (EG-PL) distribution of Mirmostafaee et al. (2019); 328) Generalized Odd HalfCauchy-G (GOHC-G) family of distributions of Cordeiro et al. (2016); 329) Generalized Kumaraswamy-G (GK-G) family of distributions of Nofal et al. (2019); 330) Alpha Power

Introduction

9

Inverted Exponential (APIE) distribution of Unal et al. (2018); 331) Alpha Power Inverse Weibull (APIW) distribution of Basheer (2019); 332) Kumarswamy Log-Logistic Weibull (KLLoGW) distribution of Mdlongwa et al. (2019); 333) Mrashall-Olkin Kappa (MOK) distribution of Javad et al. (2019); 334) Exponentiated Gumbel Exponential (EGuE) distribution of Uwadi et al. (2019); 335) Type II Topp-Leone Power Lomax (TIITLPL) distribution of Al-Marzouki et al. (2020); 336) XGamma-G (XG-G) distribution of Cordeiro et al. (2020); 337) Weibull-Inverse Lomax (WIL) distribution of Falgore et al. (2019); 338) Kumaraswamy Alpha Power Inverted Exponential (KAPIE) distribution of Zelibe et al. (2019); 339) Odd Generalized Exponential-Exponential (OGE-E) distribution of Al-Salafi and Adham (2018); 340) Topp-Leone-Lomax (TLLx) distribution of Yadav et al. (2020); 341) Beta Generalized Exponentiated Fréchet (BGEF) distribution of Badr (2019); 342) Transmuted Generalized Inverted Exponential (TGIE) distribution of Khan (2018); 343) Transmuted Generalized Power Weibull (TGPW) distribution of Khan (2018); 344) Size Biased Gamma Lindley (SBGaL) distribution of Beghriche and Zeghdoudi (2019); 345) Skew t-Distribution of Three Degrees of Freedom (St-DTDF) mentioned in Ahsanullah and Nevzorov (2019); 346) Logarithm Transformed Lomax (LTL) distribution of Nassar et al. (2018); 347) Exponentiated Generalized Power Function (EGPF) distribution of Hassan and Nassr (2019); 348) Gompertz Length Biased Exponential (Go-LBE) distribution of Maxwell et al. (2019); 349) Gumbel-Burr XII (GUBXII) distribution of Osatohanmwen et al. (2019); 350) Type II Power Topp-Leone Generated (TIIPTL-G) family of distributions of Bantan et al. (2020); 351) Beta Kumaraswamy Marshall-Olkin-G (BKwMOG) family of distributions of Handique and Chakraborty (2017); 352) Beta Generalized Marshall-Olkin Kumarswamy-G (BGMOKw-G) distribution of Handique and Chakraborty (2017); 353) Beta Marshall-Olkin Kumarswamy-G (BMOKw-G) distribution of Handique and Chakraborty (2018); 354) Zografos-Balakrishnan Burr XII (ZBBXII) distribution of Altun et al. (2018); 355) Beta Kumarswamy Burr Type X (BKBX) distribution of Madaki et al. (2018); 356) Beta Kumaraswamy Marshall-Olkin-G (BKwMO-G) family of distributions of Handique and Chakraborty (2017); 357) Poisson Burr X Burr XII (PBXBXII) distribution of Abdelkhalek (2018); 358) Transmuted Arcsine (TA) distribution of Bleed and Abdelali (2018); 359) Poisson Burr X Generalized Lomax (PBXGL) distribution of Ibrahim and Yousof (2020); 360) Transmuted Power Function (TPF) distribution of Ahsan ul Haq et al. (2016); 361) Kumaraswamy Moment Exponential (KwME) distribution of Hashim et al. (2019); 362) Marshall-Olkin Length Biased Exponential (MOLBE) distribution of Ahsan ul Haq et al. (2019); 363) Generalized Moment Exponential Power Series (GMEPS) distribution of Iqbal et al. (2020); 364) Truncated Exponential Skew Logistic (TESL) distribution of Mirzadeh and Iranmanesh (2019); 365) Balakrishnan Alpha Skew Normal2 (BASN2 ) distribution of Hazrika et al. (2019); 366) Alpha Beta Skew Logistic-G (ABSLG) distribution of Shah and Hazarika (2019); 367) Bimodal Alpha Skew LogisticG2 (BASLG2 ) distribution of Shah et al. (2019); 368) Generalized Modified Exponential-G (GMEG) family of distributions of Handique et al. (2020); 369) Doubly Truncated Extreme Value Type I (DTEVTI) distribution of Chakraborty and Sarma (2017); 370) Lomax Exponential (LE) distribution of Ijaz and Alamgir (2019); 371) Generalized Odd Log-Logistic Exponential (GOLLEx) distribution of Afify et al. (2020); 372) Extended Odd Weibull Exponential (EOWEx) distribution of Afify and Abdo Mohamed (2020); 373) New Libby-Novick (NLN) distribution of Ahmed (2020); 374) Modified Burr XII (MBXII) distribution of Jamal et al. (2019);

10

G. G. Hamedani

375) Poisson Odd Generalized Exponential (POGE) family of distributions of Muhammad (2020); 376) Modified Odd Weibull-G (MOW-G) family of distributions of Chesneau and El Achi (2020); 377) Generalized Uniform (GU) distribution of Jayakumar and Sankaran (2016); 378) McDonald Modified Burr-III (McMB-III) distribution of Mukhtar et al. (2018); 379) Generalized Lindley (GL) distribution of Hamedani (2020); 380) Weibull Marshall-Olkin Lindley (WMOL) distribution of Afify et al. (2020); 381) Generalized Inverse Marshall-Olkin (GIMO) family of distributions of Jayakumar and Sankaran (2020); 382) Poisson Burr X Inverse Rayleigh (PBX-IR) distribution of Abdelkhalek (2020); 383) Kumarswamy Reciprocal (KR) family of distributions of Bleed (202); 384) Log-Weighted Pareto (LWP) distribution of Mandouh and Mohamed (2020); 385) Alpha Power Transformed Extended Exponential (APTEE) distribution of Alghamedi et al. (2019); 386) Lomax Exponentiated Weibull (LEW) distribution of Ansari and Nofal (2020); 387) Intervened Geometric Compound (IGC) family of distributions of Jayakumar and Sankaran (2020); 388) Intervened Negative Binomial Compound (INBC) family of distributions of Jayakumar and Sankaran (2020); 389) Intervened Binomial Compound (IBC) family of distributions of Jayakumar and Sankaran (2020); 390) Intervened Poisson Compound (IPC) family of distributions of Jayakumar and Sankaran (2020); 391) Odd Log-Logistic Exponential Gaussian (OLLExGa) distribution of Vasconcelos et al. (2020); 392) Gompertz Fréchet (GFr) distribution of Oguntunde et al. (2020); 393) Weighted Exponential Gompertz (WE-G) distribution of Abd El-Bar and Ragab (2020); 394) Rayleigh Rayleigh (RR) distribution of Ateeq et al. (2020); 395) Chen-G (CG) class of distributions of Anzagra et al. (2020); 396) Power Muth (PM) distribution of Jodrá et al. (2017); 397) Cubic Transmuted Power Function (CTPF) distribution of Ansari et al. (2019); 398) Weibull Alpha Power Inverted Exponential (WAPIE) distribution of Fee-Eyefia et al. (2020); 399) Exponentiated-Epsilon (E-Epsilon) distribution of Esbond and Fuumilayo (2018); 400) Gompertz-Alpha Power Inverted Exponential (GAPIE) distribution of Eghwerido et al. (2020); 401) Gompertz Extended Generalized Exponential (G-EGE) distribution of (Eghwerido et al. (2020); 402) Extended Odd Log-Logistic (EOLL-G) family of distributions of Esmaeili et al. (2020); 403) Generalized Odd Generalized Exponential G (GOGE-G) distribution of Al-Babrain et al. (2020); 404) Slash Maxwell (SM) distribution of Acitas et al. (2020); 405) Modified T-X (MT-X) family of distributions of Aslam et al. (2020); 406) Double Truncated Transmuted Fréchet (DTTF) distribution of Iqbal et al. (2019); 407) Burr-Hatke Logarithmic BurrXII (BH-BXII) distribution of Mansour et al. (2020); 408) Odd Inverse Pareto-Exponential (OIPEx) distribution of Aldahlan and Afify (2020); 409) Poisson Rayleigh Generalized Lomax (PRGLx) distribution of Ansari et al. (2020); 410) Generalized Odd Log-Logistic-G (GOLL-G) distribution of Rezk (2020); 411) MarshallOlkin Generalized Pareto (MOGP) distribution of Ahmad and Almetwally (2020); 412) Log-Balakrishnan-Alpha-Skew-Normal (LBASN2 (α)) distribution of Shah et al. (2020); 413) Flexible Additive Weibull (FAW) distribution of Khalil et al. (2020); 414) Transmuted Weibull (TW) distribution of Pobo´ciková et al. (2018); 415) Cubic Transmuted (CT) family of distributions of Rahman et al. (2018); 416) Transmuted Modified Weibull (TMW) distribution of Khan et al. (2018); 417) Transmuted Half Normal (THN) distribution of Balaswamy (2018); 418) Cubic Rank Transmuted Fréchet (CRTF) distribution of Celik (2018); 419) Weighted Garima (WG) distribution of Eyob and Shanker (2018); 420) Exponentiated Exponential Lomax (EEL) distribution of Adeleke et al. (2019); 421) Transmuted

Introduction

11

Half Normal (THN) distribution of Abayomia and Adeleke (2019); 422) Transmuted Ishita (TI) distribution of Gharaibeh and Al-Omari (2019); 423) Transmuted Generalized Extreme Value (TGEV) distribution of Otiniano et al. (2019); 424) Transmuted Burr Type X (TBX) distribution of Khan et al. (2019); 425) Odd Lindley Exponentiated Weibull (OLi-EW) distribution of Refaie (2019); 426) Transmuted Lomax-G (TL-G) family of distributions of Hassan et al. (2020); 427) Power Lindley Geometric (PLG) distribution of Alkarni and Alshehri (2020); 428) Odd Lomax Exponential (OLxEx) distribution of Afify (2020); 429) Jamal Weibull-X (JW-X) family of distributions of Handique et al. (2020); 430) Topp-Leone Exponentiated-G (TLEx-G) family of distributions of Ibrahim et al. (2020); 431) New Extended Burr III (NEBIII) distribution of Handique et al. (2020); 432) Zero Truncated Poisson Topp Leone Fréchet (ZTPTL-Fr) distribution of Hamed (2020); 433) Alpha Power Transformed Inverse Lindley (APTIL) distribution of Dey et al. (2018); 434) Beta Type I Generalized Half Logistic (BTIGHL) distribution of Awodutire et al. (2020); 435) BurrHatke Extended Burr XII (BHEBXII) distribution of Refaie (2018); 436) Wrapped Lindley (WL) distribution of Joshi and Jose (2020); 437) Truncated Cauchy Power-G (TCP-G) family of distributions of Aldahlan et al. (2020); 438) Slash Lindley-Weibull (SLW) distribution of Reyes et al. (2018); 439) Risti´c-Balakrishnan Extended Exponential (RBEE) distribution of Gomes-Silva et al. (2018); 440) Exponentiated Generalized Standardized Gumbel (EGSGu) distribution of de Andrade et al. (2019); 441) Unit-Lindley (unit-Lindley) distribution of Mazucheli et al. (2019); 442) Odd Log-Logistic Dagum (OLLDa) distribution of Domma et al. (2018); 443) Type I Half-Logistic Modified Weibull (TIHLMW) distribution of Elbatal et al. (2019); 444) Odd Dagum (OddD-G) family of distributions of Afify and Alizadeh (2020); 445) Unit-Weibull (UW) distribution of Mazucheli et al. (2018); 446) Generalized DUS Lindley (GDUSL) distribution of Maurya et al. (2020); 447) Poisson Rayleigh Burr XII (PRBXII) distribution of Ibrahim (2020); 448) Exponential-Gamma (EG) distribution of Umar and Yahya (2020); 449) Extended Beta Power Function (EBPF) distribution of Nawaz et al. (2020); 450) Arcsine Exponentiated-X (ASE-X) family of distributions of He et al. (2020); 451) Lindley Quasi XGamma (LQXG) distribution of Hassan et al. (2020); 452) Type II Half Logistic Exponentiated Exponential (TIIHLEE) distribution of Abdulkabir and Ipinyomi (2020); 453) Weighted Ishita (WI) distribution of Hassan et al. (2020); 454) Quasi Sujatha (QS) distribution of Shanker (2016); 455) Weighted Nakagami (WN) distribution of Mudiasir and Ahmad (2017); 456) Weighted Version of Generalized Inverse Weibull (WVGIW) distribution of Mudiasir and Ahmad (2018); 457) Weighted Inverse Lévy (WIL) distribution of Ahmad et al. (2019); 458) Exponentiated Length Biased Exponential (ELBE) distribution of Maxwell et al. (2019); 459) LengthBiased Suja (LBS) distribution of AL-Omari and Alsmairan, (2019); 460) Alpha-Power Generalized Inverse Lindley (APGIL) distribution of Jan et al. (2019); 461) Exponentiated Cubic Transmuted Exponential (ECTE) distribution of Adeyinka (2020); 462) Transmuted Alpha Power Inverse Rayleigh (TAPIR) distribution of Malik and Ahmad (2019); 463) New Beta Power Transformed (NBPT) family of distributions of Ahmad et al. (2020); 464) Generalized Gamma Exponentiated Weibull (GGEW) distribution of Abid et al. (2020); 465) Marshall-Olkin Exponential Gompertz (MOEGo) distribution of Khaleel et al. (2020); 466) Exponentiated Truncated Inverse Weibull-Generated (ETIW-G) family of distributions of Almarashi et al. (2020); 467) Lindley Rayleigh (LR) distribution of Cakmakyapan and Ozel (2018); 468) Generalized Weibull Uniform (GWU) distribution of Al Abbasi et al.

12

G. G. Hamedani

(2019); 469) Burr XII Uniform (BXIIU) distribution of Nasir et al. (2018); 470) G-FixedTopp-Leone (G-FTL) class of distributions of Ali et al. (2020); 471) Generalized New Extended Weibull (GNEW) distribution of Ahmad et al. (2017); 472) New Exponential Trigonometric (NET) distribution Bakouch et al. (2018); 473) Power Log-Dagum (PLD) distribution of Bakouch et al. (2018); 474) Poly-Exponential Transformation (PET) distribution of Chesneau et al. (2018); 475) New Weighted Transmuted Exponential (NWTE) distribution of Chesneau et al (2020); 476) Kumarswamy Poisson-G (KwP-G) family of distributions of Charkraborty et al. (2020); 477) Truncated Burr-G (TB-G) class of distributions of Jamal et al. (2020); 478) Alpha Power Transformation Lomax (APTL) distribution of Dey at al. (2019); 479) Weighted Inverted Weibull (WIW) distribution of Dey et al. (2019); 480) Odd Exponential-Pareto IV (OEPIV) distribution of Baharith et al. (2020); 481) T-Dagum{Y } (TD{Y }) class of distributions of Ekum et al. (2020); 482) Generalized Odd Inverted Exponential-G (GOIE-G) family of distributions of Chesneau and Djibrila (2019); 483) Marshall-Olkin Burr Exponential-2 (MOBE-2) distribution of Al-Babtain et al. (2020); 484) Exponentiated Power Generalized Weibull Power Series (EPGWPS) family of Aldahlan et al. (2020); 485) Exponentiated Transmuted Weibull Geometric (ETWG) distribution of Al-Zahrani et al. (2015); 486) Complementary Exponential Geometric (CEG) distribution of Goldust et al. (2017); 487) Complementary Exponentiated Lomax-Poisson (CELP) distribution of Kumar et al. (2020); 488) Alpha Power Transformed Log-Logistic (APTLL) distribution of Aldahlan (2020); 489) T-Kumarswamy (T-K) family of distribution of Osatohanmwen et al. (2020); 490) Zografos Balakrishnan Power Lindley (ZB-PL) distribution of Shahid et al. (2020); 491) Poisson Exponential-G (PE-G) family of distributions of Reyad et al. (2020); 492) Perturbed Half-Normal (PHN) distribution of Mahmoudi et al. (2017); 493) Normal-Poisson (NP) distribution of Mahmoudi et al. (2017); 494) Extended Exponentiated Weibull (EEW) distribution of Mahmoudi et al. (2018); 495) New Family of Heavy Tailed (NFHT) distributions of Ahmad et al. (2020); 496) New Heavy Tailed Family of Claim (NHTFC) distributions of Ahmad et al. (2020); 497) New Beta Power Transformed (NBPT) family of distributions of Ahmad et al. (2020); 498) Transmuted Type II Generalized Logistic (TTIIGL) distribution of Adeyinka (2019); 499) Beta-Complementary Exponential Power Series (BCEPS) distribution of Mahmoudi et al. (2020); 500) ZubairG (Zubair-G) family of distributions of Ahmad (2020); 501) Generalized Class (GC) of distributions of Ahmad (2020); 502) Odd Lindley Half Logistics (OLiHL) distribution of Eliwa et al. (2020); 503) Modified Beta Generalized Linear Failure Rate (MBGLFR) distribution of Jamal et al. (2019); 504)Truncated Weibull Fréchet (TWFr) distribution of Hassan et al. (2019); 505) Beta Poisson-G (BP-G) family of distributions of Handique et al. (2020); 506) Topp-Leone Marshall-Olkin-G (TLMO-G) family of distributions of Chipepa et al. (2020); 507) Beta Modified Weibull Power Series (BMWPS) distribution of Yarmoghaddam and Samani (2019); 508) Fréchet Weibull (FW) distribution of Teamah et al. (2020); 509) Odd Lomax Fréchet (OLxF) distribution of Hamed et al. (2020); 510) Burr X Weibull (BXW) distribution of Mansour et al. (2020); 511) Beta Exponential Pareto (BEP) distribution of Aryal (2019); 512) Gompertz Flexible Weibull (GoFW) distribution of Khaleel et al. (2020); 513) Type I Half-Logistic Rayleigh (TIHLR) distribution of Al-Babtain (2020); 514) Log-Beta Modified Weibull (LBMW) distribution of Silva et al. (2020); 515) Transmuted Exponential-G (TE-G) family of distributions of Mohammed and Ugwuowo (2020); 516) New Mixture of Exponential-Gamma (NMEG) distribution of

Introduction

13

Ekhosuehi et al. (2020); 517) New Generalized Akash (NGA) distribution of Tharshan and Wijekoon (2020); 518) T-R {Y} Power Series (T-R {Y} PS) family of distributions of Osatohanmwen et al. (2020); 519) New Odd Log-Logistic Chen (NOLL-Ch) distribution of Zamani et al. (2020); 520) Transmuted Generalized Lindley (TGL) distribution of Louzada et al. (2018); 521) Beta Exponentiated Weibull Geometric (BEWG) distribution of Louzada et al. (2018); 522) Beta Exponentiated Nadarajah-Haghighi (BENH) distribution of Saboor et al. (2018); 523) Exponentiated Additive Weibull (EAW) distribution of Aljouiee et al. (2018); 524) Transmuted Kumaraswamy Lindley (TKL) distribution of Elgarhy et al. (2018); 525) Weighted Power Lindley (WPL) distribution of Rather and Ozel (2020); 526) Marshall-Olkin Extended Power Lomax (MOEPL) distribution of Gillariose and Tomy (2020); 527) Unit Modified Burr-III (UMBIII) distribution of Ahsan ul Haq et al. (2020); 528) Reflected Power Function (RPF) distribution of Zaka et al. (2020); 529) Transmuted Odd Fréchet-G (TOFr-G) family of distributions of Badr et al. (2020); 530) UnitBirnbaum-Saunders (UBS) distribution of Mazucheli et al. (2018); 531) erf-G (erf-G) family of distributions of Fernández and de Andrade (2020); 532) New Unit-Lindley (NUL) distribution of Mazucheli et al. (2020); 533) Odd Generalized Exponentiated Inverse Lomax (OGE-IL) distribution of Maxwell et al. (2019); 534) Marshall-Olkin Inverse-Lomax (MO-IL) distribution of Maxwell et al. (2019); 535) Odd Lindley-Rayleigh (OLR) distribution of Ieren et al. (2020); 536) Cubic Transmuted Gompertz (CTG) distribution of Ogunde et al. (2020); 537) Type II Topp-Leone-Power Lomax (TIITL-PL) distribution of Aryuyuen and Bodhisuwan (2020); 538) Odd Lindley Inverse Exponential (OLINEX) distribution of Ieren and Abdullahi (2020); 539) New Generalized Rayleigh (NGR) distribution of Bhat and Ahmad (2020); 540) Marshall-Olkin Power Generalized Weibull (MOPGW) distribution of Afify et al. (2020); 541) Type II Topp-Leone Power Ishita (TIITLPI-G) distribution of Ikechukwu et al. (2020); 542) Weighted Inverse Nakagami (WINK) distribution of Sarma and Das (2020); 543) Generalized Marshall-Olkin Poisson-G (GMOP-G) family of distributions of Handique et al. (2020); 544) Half-Logistic XGamma (HLXG) distribution of Bantan et al. (2020); 545) Log-Gamma-Generated (LGG1 ) family of distributions of Amini et al. (2012); 546) Log-Gamma-Generated (LGG2 ) family of distributions of Amini et al. (2012); 547) McDonald Gumbel (MG) distribution of de Brito et al. (2016); 548) Alpha-Beta Skew Logistic G (ABSLG) distribution of Esmaeili et al. (2020); 549) Generalized Transmuted Poisson-G (GTPG) family of distributions of Mallick and Ghosh (2018); 550) Generalized Marshall-Olkin Transmuted-G (GMOT-G) family of distributions of Handique et al. (2020); 551) Generalized Odd Linear Exponential (GOLE) family of distributions of Marzouk et al. (2020); 552) Exponentiated Odd Chen-G (EOCh-G) family of distributions of Eliwa et el. (2020); 553) Transmuted Complementary Exponential Power (TCEP) distribution of Tanis et al. (2020); 554) Hyperbolic Cosine Weibull (HCW) distribution of Kharazmi et al. (2019); 555) Burr X Exponential-G (BXE-G) family of distributions of Sanusi et al. (2020); 556) Extended Generalized Inverse Exponential (EGIEx) distribution of Ibrahim et al. (2020); 557) Weibull-Negative Binomial (WNB) distribution of Korkmaz et al. (2011); 558) Unit Generalized Half Normal (UGHN) distribution of Korkmaz (2020); 559) Generalization of Exponential and Lindley (GEL) distributions of Yazar and Korkmaz (2016); 560) Unit Nadarajah-Haghighi Generated (UNH-G) family of distributions of Nasiru et al. (2020); 561) Lomax D function Generalized Weibull (LDGW) distribution of Hussain et al. (2020); 562) Right Truncated Power Lomax (RTPL) distribution

14

G. G. Hamedani

of Hassan et al. (2020); 563) Exponentiated Garima (EG) distribution of Rather and Subramanian (2020); 564) New Extended Weibull (NEW) family of distributions of Zichuan et al. (2020); 565) Kumarswamy Sushila (KwS) distribution of Shawki and Elgarhy (2017); 566) Alpha Power Transformed Inverse Lomax (APTIL) distribution of ZeinEldin et al. (2020); 567) New Extended-F (NE-F) family of distributions of Khosa et al. (2020); 568) Topp-Leone Rayleigh (TLR) distribution of Olayode (2019); 569) Marshall-Olkin Topp Leone-G (MOTL-G) family of distributions of Khaleel et al. (2020); 570) Right TruncatedX (RT-X) family of distributions of Alzaatreh et al. (2020); 571) Generalized MarshallOlkin Inverse Lindley (GMOIL) distribution of Bantan et al. (2020); 572) Generalized Reciprocal Exponential (GRE) distribution of Mansour et al. (2020); 573) Generalized Transmuted Moment Exponential (GTME) distribution of Hassan et al. (2020); 574) Generalized Weighted Exponential (GWEx) distribution of Abbas et al. (2020); 575) Exponentiated Transmuted Length-Biased Exponential (ETLBE) distribution of Abbas et al. (2020); 576) Gamma Inverse Weibull (GIW) distribution of Abbas et al. (2019); 577) Transmuted Alpha Power-G (TAPO-G) family of distributions of Eghwerido et al. (2020); 578) Exponentiated Poisson-Exponential (EPE) distribution of Louzada et al. (2020); 579) Exponentiated Power Function (EPF) distribution of Arshad et al. (2020); 580) Alpha Logarithmic Transformed Weibull (ALTW) distribution of Nassar et al. (2018); 581) Odd Exponentiated Half-Logistic Exponential (OEHLEx) distribution of Afify et al. (2018); 582) Odd Burr III Exponential (OBIIIE) distribution of Umar et al. (2019); 583) Rayleigh-Geometric (RG) distribution of Okorie et al. (2019); 584) Weibull Exponentiated Exponential (WEE) distribution of Usman et al. (2020); 585) Logistic Exponential (LE) distribution of Ali et al. (2020); 586) Generalized Gamma-G (GG-G) family of distributions of Nasir et al. (2020); 587) Exponentiated Shanker (E-Sh) distribution of Abdollahi et al. (2019); 588) Extended Generalized Lindley (EGL) distribution of Ranjbar et al. (2019); 589) RisticBalakrishnan Odd Log-Logistic-G (RBOLL-G) family of distributions of Esmaeili et al. (2020); 590) Burr XII Exponentiated Exponential (BrXIIEE) distribution of Ibrahim et al. (2020); 591) Mixture Pareto Log-Gamma (MPLG) distribution of Bhati et al. (2019); 592) Exponentiated Two Parameter Pranav (ETPP) distribution of Hassan et al. (2020); 593) New Lifetime Exponential-Weibull (NLTE-W) distribution of Huo et al. (2020); 594) New Cubic Rank Transmutation (NCRT) distribution of Hameldarbandi and Yilmaz (2020); 595) Alpha Power Exponentiated Exponential (APExE) distribution of Afify et al. (2020); 596) Odd Fréchet Inverse Lomax (OFIL) distribution of ZeiEldin et al. (2020); 597) Topp-Leone Power Lindley (TLPL) distribution of Opone et al. (2020); 598) Gull Alpha Power (GAP) family of distributions of Ijaz et al. (2020); 599) Modi Generator (MG) distribution of Modi et al. (2020); 600) Cubic Transmuted Exponentiated Pareto-1 (CTEP-1-G) distribution of Eledum (2020); 601) Cubic Transmuted Exponentiated Pareto-1 (CTEP-1-R) distribution of Eledum (2020); 602) Slashed Quasi-Gamma (SQG) distribution of Iriate et al. (2020); 603) Log-Epsilon-Skew Normal (LESN) distribution of Hutson et al. (2020); 604) Generalized Odd Log-Logistic Log-Normal (GOLLLN) distribution of Vasconcelos et al. (2020); 605) Slash Power Maxwell (SPM) distribution of Segovia et al. (2020); 606) Modified Slashed Half-Normal (MSHN) distribution of Olmos et al. (2019); 607) One Parameter Polynomial Exponential-G (OPPE-G) family of distributions of Maiti and Pramanik (2020); 608) Beta Burr Type X (BBX) distribution of Ibrahim and Khaleel (2020); 609) Gamma Burr Type X (GBX) distribution of Ibrahim and Khaleel (2020); 610) Weibull Burr

Introduction

15

Type X (WBX) distribution of Ibrahim and Khaleel (2020); 611) Extended Odd WeibullG (ExOW-G) distribution of Afify and Mohamed (2020); 612) Sinh Inverted Exponential (SIE) distribution of Hemeda and Abdallah (2020); 613) Exponential Skew-Normal (ESN) distribution of Martinez-Florez et al. (2020); 614) T-Burr (T-Burr{Y }) family of distributions of Nasir et al. (2017); 615) Transmuted General (T-G) family of distributions of Bakouch et al. (2017); 616) Odd Gamma Weibull-Geometric (OGWG) distribution of Arshad et al. (2019); 617) Generalized Inverted Kumarswamy Generated (GIKw-G) family of distributions of Jamal et al. (2019); 618) Gamma Kumarswamy-G (GKw-G) family of distributions of Arshad et al. (2019); 619) Weibull Burr XII (WBXII) distribution of Nasir et al. (2018); 620) Odd Generalized Gamma-G (OGG-G or GG-G) family of distributions of Nasir et al. (2020); 621) Marshall-Olkin Odd Burr III-G (MOOB-G) family of distributions of Afify et al. (2019); 622) Gamma Power Half-Logistic (GPHL) distribution of Arshad et al. (2019); 623) Topp-Leone Weibull-Lomax (TLWLx) distribution of Jamal et al. (2019); 624) Minimum Weibull-Burr (minWB) distribution of Nasir et al. (2019); 625) Box-Cox Gamma-G (BCG-G) family of distributions of Jamal and Chesneau (2019); 626) Modified Beta Generalized Linear Failure Rate (MBGLFR) distribution of Elbatal et al. (2019); 627) New Modified Burr III (NMBIII) distribution of Jamal et al. (2018); 628) Inverted Modified Lindley (IML) distribution of Chesneau et al. (2020); 629) Type II General Inverse Exponential (TIIGIE) family of distributions of Jamal et al. (2020); 630) Exponentiated Half-Logistic Lomax (EHLLx) distribution of Jamal et al. (2019); 631) Generalized Gamma-Generalized Inverse Weibull (GG-GIW) distribution of Abid et al. (2018); 632) Log-Weighted Exponential (log-WE) distribution of Altun (2020); 633) Generalized Raised Cosine (GENRC) distribution of Ahsanullah et al. (2019); 634) Sine Kumarswamy-G (SKG) family of distributions of Chesneau and Jamal (2019); 635) Extended Exp-G (EE-G) family of distributions of Alizadeh et al. (2020); 636) Weighted Exponential (WE) distribution of Altun (2029); 637) Lindley Negative-Binomial (LNB) distribution of Mansoor et al. (2020); 638) Marshall-Olkin Transmuted-G (MOT-G) family of distributions of Afify et al. (2020); 639) Ratio of Two Independent Weibull and Lindley (RTIWL) distribution of Hassan et al. (2020); 640) Product of Two Independent Weibull and Lindley (PTIWL) distribution of Hassan et al. (2020); 641) Modified Kies Generalized (MKi-G) family of distributions of Al-Babtain et al. (2020); 642) Ratio Exponentiated General (RE-G) family of distributions of Bantan et al. (2020); 643) Gompertz Exponential (GoEp) distribution of Ademola and Sunday (2020); 644) Minimum Gumbel Burr (minGuBu) distribution of Jamal et al. (2020); 645) Transmuted Power Gompertz (TPG) distribution of Eraikhuemen et al. (2020); 646) Burr X-G (BX-G) family of distributions of Sanusi et al. (2020); 647) Hamza (Hamza) distribution of Aijaz et al. (2020); 648) Inverse Lomax-G (IL-G) family of distributions of Falgore and Doguwa (2020); 649) Zubair-Inverse Lomax (ZIL) distribution of Falgore (2020); 650) Skew Scale Mixtures Normal (SSMN) distribution of Ferreira et al. (2020); 651) Akash (Akash) distribution of Shanker (2015); 652) A Generalization of Sujatha (AGS) distribution of Shanker et al. (2017); 653) Two-Parameter Sujatha (TPS) distribution of Tesfay and Shanker (2018); 654) New Two-Parameter Sujatha (NTPS) distribution of Tesfay and Shanker (2018); 655) Another Two-Parameter Sujatha (ATPS) distribution of Tesfay and Shanker (2019); 656) Lomax Inverse Weibull (LxIW) distribution of Goual et al. (2020); 657) Odd Burr Generalized Rayleigh (OBGR) distribution of Ali et al. (2020); 658) Marshall-Olkin Lehmann Burr X (MOLBX) distribution of Ali et al.

G. G. Hamedani

16

(2020); 659) Transmuted Topp-Leone Weibull (TTL-W) distribution of Ibrahim and Yousof (2020); 660) Burr X Generalized Burr XII (BXGBXII) distribution of Elsayed and Yousof (2020); 661) Generalized Odd Generalized Exponential Fréchet (GOGEFr) distribution of Elsayed and Yousof (2020); 662) Topp-Leone Lindley (TLLi) distribution of Sharma et al. (2020); 663) Logarithmic Transformed Inverse Weibull (LTIW) distribution of Afify et al. (2020); 664) Kumarswamy Alpha Power-G (KAP-G) family of Distributions of Mead et al. (2020); 665) Flexible Weibull Burr XII (FWBXII) distribution of Kamal and Ismail (2020); 666) Extended Poisson Lomax (EPLx) distribution of Hamed (2020); 667) Unit Johnson SU (UJSU ) distribution of G˝und˝uz and Korkmaz (2020);. These characterizations are presented in different directions: (i) based on the ratio of two truncated moments; (ii) in terms of the hazard function; (iii) in terms of the reverse (reversed) hazard function and (iv) based on the conditional expectation of certain function of the random variable. Note that (i) can be employed also when the cd f (cumulative distribution function) does not have a closed form. In defining the above distributions we shall try to employ the same parameter notation as used by the original authors.

1.1 Preliminaries We follow the same order as listed above.

1.1.1

Extended Weighted Exponential (EWE)

The cd f and pd f (probability density function) of EWE are given, respectively, by F (x; α, β, λ) = 1 −C0 (α, β, λ)e−λx +C1 (α, β, λ)e−λ(α+1)x + C2 (α, β, λ)xe−λ(α+1)x ,

(1.1.1)

x ≥ 0, and f (x; α, β, λ) = x > 0 , where C1 (α, β, λ) =

1.1.2

 λ (1 + α)2 e−λx  λ + β − (λ + β + αβλx) e−αλx , α (λ (1 + α) + αβ)

α, β, λ are all positive parameters and C0 (α, β, λ) =

(1.1.2) (1+α)2 (λ+β) α(λ(1+α)+αβ) ,

(1+α)(λ+β)+αβ (1+α)λβ α(λ(1+α)+αβ) , C2 (α, β, λ) = (λ(1+α)+αβ) .

Exponentiated Generalized Extended Pareto (EGEP)

The cd f and pd f of EGEP are given, respectively, by

and

h i a b F (x; a, b, η) = 1 − (1 + ηx)− η , x ≥ 0, f (x; a, b, η) = ab (1 + ηx)−

where a, b, η are all positive parameters.

(a+η) η

h

a

1 − (1 + ηx)− η

ib−1

(1.1.3) , x > 0,

(1.1.4)

Introduction

17

Remark 1.1. Gómez et al. (2014) proposed the following distribution h iα F (x; α, θ, λ) = 1 − (1 + λx)−θ , x ≥ 0,

which is the same as cd f (1.1.3). Furthermore, a generalization of Gómez et al. was introduced by Mead (2015) which has been characterized in Hamedani and Maadooliat (2017).

1.1.3

Inverse Weibull Generator (IWG)

The cd f and pd f of IWG are given, respectively, by (  −β ) G (x; η) F (x; β, θ, η) = exp −θβ , x ∈ R, G (x; η)

(1.1.5)

and βθβ g (x; η) [G (x; η)]−β−1 ×  −β+1 G (x; η) (  −β ) G (x; η) exp −θβ , x ∈ R, G (x; η)

f (x; β, θ, η) =

(1.1.6)

where β, θ are positive parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η), which may depend on the parameter vector η.

1.1.4

Generalized Burr X-G (GBX-G)

The cd f and pd f of GBX-G are given, respectively, by F (x; α, β, ϕ) =

"  2 #)β G (x; ϕ)α 1 − exp − , x ∈ R, 1 − G (x; ϕ)α

(

(1.1.7)

and "  2 # 2αβg (x; ϕ) G (x; ϕ)2α−1 G (x; ϕ)α f (x; α, β, ϕ) = exp − ×  3 1 − G (x; ϕ)α 1 − G (x; ϕ)α ( "  2 #)β−1 G (x; ϕ)α 1 − exp − , 1 − G (x; ϕ)α

(1.1.8)

x ∈ R, where α, β are positive parameters and G (x; ϕ) is a baseline cd f with the corresponding pd f g (x; ϕ), which may depend on the parameter vector ϕ.

18

G. G. Hamedani

Remark 1.2. Hassan and Elgarhy (2019) proposed the following distribution "  #a  −α

F (x; α, β, a, η) = 1 − e

K(x;η) 1−K(x;η)

β

, x ∈ R.

Taking K (x; η) = G (x; ϕ)α , F (x; α, β, a, η) provides a more general distribution than (1.1.7). The cd f F (x; α, β, a, η) has been characterized in Hamedani (2019).

1.1.5

Generalized Lindley Power Series (GLPS)

The cd f and pd f of GLPS are given, respectively, by h h   iα i  λx C 1 − 1 − 1 + λ+1 e−λx θ F (x; α, θ, λ) = 1 − , x ≥ 0, C (θ)

(1.1.9)

and    α−1 αλ2 θe−λx (1 + x) λx −λx f (x; α, θ, λ) = 1− 1+ e × (λ + 1)C (θ) λ+1  α      λx 0 −λx C 1− 1− 1+ e θ , λ+1

(1.1.10)

n 0 x > 0, where α, θ, λ are all positive parameters and C (θ) = ∑∞ n=1 an θ , an s are non-negative real numbers depending on n such that C (θ) is finite.

Remark 1.3. Alizadeh et al. (2017) introduced the following distribution       β α λxβ C θ − θ 1 − 1 + λ+1 e−λx F (x; α, β, θ, λ) = 1 − , x ≥ 0, C (θ) which seems to be a more general version of (1.1.9). The cd f F (x; α, β, θ, λ) has been characterized in Hamedani (2018b).

1.1.6

Transmuted Exponentiated Additive Weibull (TEAW)

The cd f and pd f of TEAW are given, respectively, by   θ β δ F (x; α, β, γ, θ, δ, λ) = 1 − e−αx −γx ×   δ  −αxθ −γxβ , x ≥ 0, 1+λ−λ 1−e and

(1.1.11)

Introduction

19

    θ β θ β δ−1 f (x; α, β, γ, θ, δ, λ) = δ αθxθ−1 + γβxβ−1 e−αx −γx 1 − e−αx −γx   δ  −αxθ −γxβ × 1 + λ − 2λ 1 − e , (1.1.12)

x > 0, where α, β, γ, θ, δ all positive and λ (|λ| ≤ 1) are parameters.

Remark 1.4. Afify et al. (2017) proposed the following distribution F (x; α, β, γ, a, b)  h  ia h  i−a b = 1 − 1 − αa 1 − exp − (γx)β α + (1 − α) exp − (γx)β , x ≥ 0,

where α (0 < α < 1) and β, γ, a, b all positive, are parameters. Characterizations of cd f (1.1.11) will be similar to those of F (x; α, β, γ, a, b) which are given in Hamedani (2018b).

1.1.7

Exponentiated Weighted Exponential (EWE)

The cd f and pd f of EWE are given, respectively, by

F (x; α, θ, λ) = and



 θ 1  α+1 −λx −λ(1+α)x 1−e − 1−e , x ≥ 0, α 1+α

 α + 1 −λx  −λαx f (x; α, θ, λ) = θ λe 1−e × α   θ−1 α+1 1  1 − e−λx − 1 − e−λ(1+α)x , α 1+α

(1.1.13)



(1.1.14)

x > 0, where α, θ, λ are all positive parameters.

1.1.8

New Weighted Exponential (NWE)

The cd f and pd f of NWE are given, respectively, by

and

F (x; α, λ) = 1 − e−α(1+λ)x , x ≥ 0,

(1.1.15)

f (x; α, λ) = α(1 + λ)e−α(1+λ)x , x > 0,

(1.1.16)

where α, λ are positive parameters. Remark 1.5. The cd f (1.1.15) is JUST an exponential distribution with parameter α(1 + λ), which has been characterized by many authors including I.

G. G. Hamedani

20

1.1.9

Gompertz Lomax (GoLom)

The cd f and pd f of GoLom are given, respectively, by αγ θ F (x; α, β, γ, θ) = 1 − e γ {1−[1+βx] } , x ≥ 0,

(1.1.17)

αγ θ f (x; α, β, γ, θ) = θαβ (1 + βx)αγ−1 e γ {1−[1+βx] }, x > 0,

(1.1.18)

and

where α, β, γ, θ are all positive parameters. Remark 1.6. Alizadeh et al. (2017) proposed the following distribution F (x; θ, γ, η) = 1 − e

θ γ

n o −γ 1−[G(x;η)]

, x ≥ 0.

Taking G (x; η) = [1 + βx] −α , x ≥ 0 , we arrive at cd f (1.1.17). Clearly F (x; θ, γ, η) is more general than (1.1.17), which has been characterized in Hamedani (2018a).

1.1.10

Chen’s Two-Parameter Exponential Power Life-Testing (CTPEPLT)

The cd f and pd f of CTPEPLT are given, respectively, by   k λ 1−ex

F (x; λ, k) = 1 − e

, x ≥ 0,

(1.1.19)

and   xk k−1 xk λ 1−e

f (x; λ, k) = λkx

e e

, x > 0,

(1.1.20)

where λ, k are positive parameters. Remark 1.7. El-Damcese et al. (2016) introduced the following distribution h n h  d ioiθ F (x; a, b, c, d, θ) = 1 − exp − axb ecx − 1 , x ≥ 0.

Clearly F (x; a, b, c, d, θ) is more general than (1.1.19), which has been characterized in Hamedani and Najibi (2016).

1.1.11

Inverted Weighted Exponential (IWE)

The cd f and pd f of IWE are given, respectively, by F (x; α, β) = and

 αβ 1 −β  e x α + 1 − e− x , x ≥ 0, α

 β (α + 1) − β  − αβ x x e 1 − e , x > 0, αx2 where α, β are positive parameters. f (x; α, β) =

(1.1.21)

(1.1.22)

Introduction

1.1.12

21

Kumaraswamy Marshall-Oklin Exponential (KMOE)

The cd f and pd f of KMOE are given, respectively, by (

"

1 − e−λx F (x; a, b, p, λ) = 1 − 1 − 1 − pe−λx

# a )b

, x ≥ 0,

(1.1.23)

and
a−1 abλ (1 − p) e−λx 1 − e−λx f (x; a, b, p, λ) = ×
a+1 1 − pe−λx ( " #a )b−1 1 − e−λx 1− , x > 0, 1 − pe−λx

(1.1.24)

where a, b, λ all positive and p < 1 are parameters. Remark 1.8. Rasekhi et al. (2018) proposed the following distribution c

F (x; a, b, c, λ, p) = 1 − 1 −

1 − e−(λx) c 1 + pe−(λx)

!a !b

, x ≥ 0.

The characterizations of KMOE will be similar to those of cd f F (x; a, b, c, λ, p) which have been presented in the Rasekhi et al.’s paper.

1.1.13

Kumaraswamy Weibull (KumW)

The cd f and pd f of KumW are given, respectively, by

and

 a b F (x; a, b, c, λ) = 1 − 1 − [1 − exp {− (λx)c}] , x ≥ 0, f (x; a, b, c, λ) = abcλc xc−1 exp {− (λx)c } [1 − exp {− (λx)c}]  a b−1 1 − [1 − exp {− (λx)c }] ,

a−1

(1.1.25)

× (1.1.26)

x > 0, where a, b, c, λ are all positive parameters.

Remark 1.9. Taking p = 0 in the cd f F (x; a, b, c, λ, p) of Remark 1.8 , we arrive at (1.1.25).

G. G. Hamedani

22

1.1.14

Kumaraswamy Half-Logistics (KH-L)

The cd f and pd f of KH-L are given, respectively, by (

"

1 − e−λx F (x; a, b, λ) = 1 − 1 − 1 + e−λx

#a )b

(1.1.27)

,

and " #a−1 2abλe−λx 1 − e−λx f (x; a, b, c, λ) = ×
2 1 + e−λx 1 + e−λx ( " #a )b−1 1 − e−λx 1− , x > 0, 1 + e−λx

(1.1.28)

where a, b, λ are all positive parameters. Remark 1.10. Taking p = −1 in the cd f (1.1.23) , we arrive at (1.1.27).

1.1.15

Transmuted Generalized Linear Exponential (TGLE)

The cd f and pd f of TGLE are given, respectively, by "  #"  − αx+ β2 x2

θ

F (x; α, β, θ, λ) = 1 − e

 θ # − αx+ β2 x2

1 + λe

,

(1.1.29)

x ≥ 0, and   θ β 2 θ−1 − αx+ β2 x2 f (α, β, θ, λ) = θ (α + βx) αx + x e × 2 "   # 

− αx+ 2β x2

1 − λ + 2λe

θ

,

(1.1.30)

x > 0, where α, β, θ all positive and λ (|λ| ≤ 1) are parameters. Remark 1.11. Nofal et al. (2017) proposed the following distribution h iδ  h iδ  −(αxθ +γxβ ) −(αxθ +γxβ ) F (x; α, β, γ, δ, λ) = 1 − e 1+λ−λ 1−e .

Characterizations of (1.1.29) are similar to those of F (x; α, β, γ, δ, λ) which have appeared in Hamedani (2017).

Introduction

1.1.16

23

Kumaraswamy-Chen (Kw-Chen), Kumaraswamy-XTG (Kw-XTG) and Kumaraswamy Flexible Weibull (Kw-FW)

The cd f 0 s of Kw-Chen, Kw-XTG and Kw-FW are given, respectively, by h  n h ioa ib β F (x; β1 , λ1 , a, b) = 1 − 1 − 1 − exp λ1 1 − ex 1 , x ≥ 0, "

(

"



F (x; β1 , λ1 , a, b) = 1 − 1 − 1 − exp λ2 α2 1 − e

x α2

β2 #)!a #b

, x ≥ 0,

(1.1.31)

(1.1.32)

and n oa b β α3 x− x3 F (x; α3 , β3 , a, b) = 1 − 1 − 1 − exp −e , x ≥ 0, 



(1.1.33)

where α1 , α2 , β1 , β2 , β3 , λ1 , λ2 , a, b are all positive parameters.

Remark 1.12. Rezaei et al. (2016) introduced the following distribution n o b θ F (x; a, b, θ, η) = 1 − 1 − {1 − [1 − K (x; η)]a } , x ∈ R.

(1.1.34)

where a, b, θ are all positive parameters and K (x; η) is the cd f of the baseline distribution which may depend on the parameter vector η. ( "  β2 #) n h io x β1 λ1 λ2 α2 x and = 1 − Taking K (x; η) = 1 − exp a 1 − e , = 1 − exp 1 − e α2 a n h io β3 exp − 1a 1 − eα3 x− x in (34), we arrive at (1.1.31), (1.1.32) and (1.1.33), respectively. Mahmoud et al. (2015), proposed the following distribution

n h
β ia ob F (x; α, β, a, b, η) = 1 − 1 − 1 − 1 − (K (x; η))α , x ∈ R,

which is a more general form of (1.1.34). Rezaei et al.’s paper was submitted before the above mentioned paper appeared and we have gathered that these papers were done independently. The reason we mention the above paper, is that it has been characterized in Hamedani’s upcoming Monograph, so we will not give a characterization of F (x; a, b, θ, η) here. We should also mention that Kw-Chen, Kw-XTG and Kw-FW distributions were published in (2014) which have priority over Rezaei et al. and Mahmoud et al.’s distributions.

1.1.17

Exponentiated Generalized Inverted Exponential (EGIE)

The cd f and pd f of EGIE are given, respectively, by h n o i −1 α β F (x; α, β) = 1 − 1 − e−x , x ≥ 0,

(1.1.35)

G. G. Hamedani

24 and

−1 oα−1 h n oα iβ−1 αβe−x n −x−1 −x−1 f (x; α, β) = 1 − e 1 − 1 − e , x > 0, x2

(1.1.36)

where α, β are positive parameters.

Remark 1.13. Mansour et al. (2018) proposed the following distribution     a b λ α −( σx ) F (x; α, σ, λ, a, b) = 1 − 1 − 1 − 1 − e , x ≥ 0. Taking b = 1 in F (x; α, σ, λ, a, b), we arrive at a more general distribution than (1.1.35). Mansour et al. distribution has been characterized in Hamedani (2019).

1.1.18

Exponentiated Generalized Exponentiated Exponential (EGEE)

The cd f and pd f of EGEE are given, respectively, by

and

   k α β − θx F (x; α, β, θ, k) = 1 − 1 − 1 − e , x ≥ 0,

 n o x k−1 αβk − x  −1 α−1 f (x; α, β, θ, k) = e θ 1 − e− θ 1 − e−x × θ    k α β−1 − θx 1− 1− 1−e ,

(1.1.37)

(1.1.38)

x > 0, where α, β, θ, k are all positive parameters.

Remark 1.14. Rodrigues and Silva (2015) proposed the following cd f   h
iλ θ −αx β F (x; α, β, λ, θ) = 1 − 1 − 1 − e , x ≥ 0,

which is the same as (1.1.37). the distribution F (x; α, β, λ, θ) has been characterized in Hamedani (2019).

1.1.19

Transmuted Exponentiated U-quadratic (TEUq)

The cd f and pd f of TEUq are given, respectively, by α  θ 3 3 F (x; α, β, θ, λ, a, b) = (1 + λ) (x − β) + (β − a) 3 α  2θ −λ (x − β)3 + (β − a)3 , 3

(1.1.39)

Introduction

25

x ∈ [a, b] and  θ−1 f (x; α, β, θ, λ, a, b) = θαθ31−θ (x − β)2 (x − β)3 + (β − a)3 ×   α  θ (x − β)3 + (β − a)3 , (1 + λ) − 2λ 3 x ∈ (a, b), where θ > 0, |λ| ≤ 1, a ∈ R, b > a, α =

1.1.20

12 (b−a)3

and β =

a+b 2

(1.1.40)

are parameters.

Generalized Gamma Burr III (GGBIII)

The cd f and pd f of GGBIII are given, respectively, by     γ α, − log 1 − 1 + F (x; α, β, θ, δ, λ, p) = Γ (α)


−β x −δ λ

 p 

,

(1.1.41)

x ≥ 0, and     
−β p α−1 x −δ pβδλ x − log 1 − 1 + λ    f (x; α, β, δ, λ, p) = 
β+1
−β x −δ x −δ Γ (α) 1 + λ 1− 1+ λ ( " !# )   x −δ −β p × exp − − log 1 − 1 + , (1.1.42) λ δ −δ−1

x > 0, where α, β, δ, λ, p are all positive parameters and γ (α, x) =

1.1.21

R x α−1 −u e du. 0 u

Beta Skew-t (BST)

The cd f and pd f of BST are given, respectively, by 1 F (x; a, b, λ, r) = B (a, b)

Z G(x;λ,r) 0

za−1 (1 − z)b−1 dz, x ∈ R,

(1.1.43)

and

f (x; a, b, λ, r) =

1 g (x; λ, r)G (x; λ, r)a−1 × B (a, b)

(1 − G (x; λ, r))b−1 , x ∈ R,

(1.1.44)

where a, b, r all positive and λ ∈ R are parameters and G (x; λ, r), g (x; λ, r) are cd f and pd f of skew t-distribution.

G. G. Hamedani

26

Remark 1.15. The Generalized Class of distributions with the following cd f was introduced in (2015) 1 F (x; α, β) = B (a, b)

Z Q(x) 0

t α−1 (1 − t)β−1 dz, x ∈ R,

where Q (x) is a cd f . This is similar to (1.1.43). The distribution F (x; α, β) has been characterized in Hamedani (2016).

1.1.22

A Class of Lindley and Weibull (ACLW)

The cd f and pd f of ACLW are given, respectively, by   βθ F (x; θ, β, η) = 1 − 1 + H (x; η) e−θH(x;η) , x ≥ 0, θ+β

(1.1.45)

and

f (x; θ, β, η) =

θ2 h (x; η) (1 + βH (x; η)) e−θH(x;η) , x > 0, θ+β

(1.1.46)

where θ, η > 0, β ≥ 0 are parameters and H (x; η) is a non-negative monotonically increasing function which depends on a parameter η > 0 and h (x; η) is the derivation of H (x; η). Remark 1.16. It should have been mention that H (x; η) may not be differentiable at a countable points of (0, ∞) for which h (x; η) will be define equal to zero.

1.1.23

Generalized Inverted Kumaraswamy (GIKum)

The cd f and pd f of GIKum are given, respectively, by  α −β , x ≥ 0, F (x; α, β, γ) = 1 − (1 + xγ )

(1.1.47)

and

 α−1 f (x; α, β, γ) = αβγxγ−1 (1 + xγ )−β−1 1 − (1 + xγ )−β , x > 0,

(1.1.48)

where α, β, γ are all positive parameters.

Remark 1.17. h The followingi cd f s have already appeared in the literature: a) α F (x; α, θ, λ) = 1 − (1 + λx)−θ , x ≥ 0 ; b) F (x; α, β) = 1 − (1 + xα )−β , x ≥ 0; c) h
−λ iα F (x; α, θ, λ) = 1 − 1 + ex/θ , x ∈ R. The characterizations of (1.1.47) are similar to those of cd f s given in a) − c). The distributions proposed via a) − c) have been characterized in Hamedani (2018a, 2018b and 2019).

Introduction

1.1.24

27

Exponentiated Generalized Extended Pareto (EGEP)

The cd f and pd f of EGEP are given, respectively, by  b F (x; a, b, η) = 1 − (1 + ηx)−a/η , x ≥ 0,

and

f (x; a, b, η) = ab (1 + ηx)−

(a+η) η

where a, b, η are all positive parameters.

 b−1 1 − (1 + ηx)−a/η , x > 0,

(1.1.49)

(1.1.50)

Remark 1.18. Clearly (1.1.49) is a special case of (1.1.47) given above.

1.1.25

Benktander Type II (BType II)

The cd f and pd f of BType II are given, respectively, by a b F (x; a, b) = 1 − xb−1 e b (1−x ), x ≥ 1,

(1.1.51)

and   a b f (x; a, b) = xb−2 axb − b + 1 e b (1−x ) , x > 1,

(1.1.52)

where a > 0, 0 < b ≤ 1 are parameters.

1.1.26

Generalized Transmuted Fréchet (GTFr)

The cd f and pd f of GTFr are given, respectively, by         α β α β F (x; α, β, λ, a, b) = exp −a 1 + λ − λ exp −b , x ≥ 0, x x

(1.1.53)

and     α β f (x; α, β, λ, a, b) = βα x exp −a x      α β × a (1 + λ) − λ (a + b) exp −b , x > 0, x β −β−1

where α, β, a, b all positive and λ (|λ| ≤ 1) are parameters. Remark 1.19. Khan et al. (2017) proposed the following distribution

(1.1.54)

G. G. Hamedani

28

  " (  β )#φ   1 α × F (x; α, β, γ, φ, λ) = 1 − 1 − exp − − γ   x x   " (  β )#φ   α 1 1 + λ 1 − exp − − γ , x ≥ 0.   x x

Clearly F (x; α, β, γ, φ, λ) is more general than cd f (1.1.53). The distribution F (x; α, β, γ, φ, λ) has been characterized in Hamedani (2018a). We would like to mention that the cd f (1.1.53) (and in turn F (x; α, β, γ, φ, λ)) is a first step generalization of the Transmuted Fréchet distribution of Mahmoud and Mandouh (2013).

1.1.27

Beta Linear Failure Rate Power Series (BLFRPS)

The cd f and pd f of BLFRPS are given, respectively, by

and

F (x; α, β, θ, a, b) = I

1−

C(θe−z ) C(θ)

 (a, b),

x ≥ 0,

θ (α + βx) e−zC0 (θe−z) × B (a, b)C (θ)  a−1  b−1 C (θe−z ) C (θe−z ) 1− , x > 0, C (θ) C (θ)

(1.1.55)

f (x; α, β, θ, a, b) =

(1.1.56)

where α, β, a, b all positive and θ ∈ (0, s) are parameters, z = αx + β x2 , B (a, b) =

2 R 1 a−1 n (1 − t)b−1 dt and C (θ) = ∑∞ n=1 an θ < ∞ ,where an ≥ 0 depends only on n. 0 t

1.1.28

Odd Lomax-G (OLxG)

The cd f and pd f of OLxG are given, respectively, by  −α G (x; ϕ) F (x; α, β, ϕ) = 1 − β β + 1 − G (x; ϕ)   −α 1 G (x; ϕ) = 1− 1+ , x ∈ R, β 1 − G (x; ϕ) α

and

  −α−1 1 G (x; ϕ) f (x; α, β, ϕ) = 1+ , x ∈ R, β 1 − G (x; ϕ) [1 − G (x; ϕ)]2 αβ−1 g (x; ϕ)

(1.1.57)

(1.1.58)

where α, β positive parameters and G (x; ϕ) , g (x; ϕ) are cd f and pd f of the baseline distribution which may depend on the parameter vector ϕ .

Introduction

29

Remark 1.20. Alizadeh et al. (2019) proposed the following distribution G (x; η) F (x; α, β, η) = 1 − 1 + β G (x; η) 



α − β1

,

x ∈ R,

which is clearly more general than cd f (1.1.57). The cd f F (x; α, β, η) has been characterized in Hamedani (2019).

1.1.29

Exponentiated Power Generalized Weibull (EPGW)

The cd f and pd f of EPGW are given, respectively, by

and

  β F (x; α, β, γ, λ) = 1 − exp 1 − (1 + λxγ )α , x ≥ 0, γ α−1 exp γ−1 (1 + λx )

f (x; α, β, γ, λ) = αβγλx

1 − (1 + λxγ )α   1−β , x > 0, 1 − exp 1 − (1 + λxγ )α 

(1.1.59)

(1.1.60)

where α, β, γ, λ are all positive parameters.

Remark 1.21. Rezaei et al. (2016) proposed the following distribution n

oθ a b

F (x; a, b, θ, ξ) = 1 − 1 − {1 − (1 − G (x; ξ)) } , x ∈ R. 1   Taking θ = 1 and G (x; ξ) = 1 − exp a 1 − (1 + λxγ )α in the above cd f , we arrive at cd f (1.1.59). The cd f F (x; a, b, θ, ξ) has been characterized in Hamedani (2017).

1.1.30

Lindley Weibull (LiW)

The cd f and pd f of LiW are given, respectively, by   h i θ β F (x; θ, α, β) = 1 − 1 + (αx) exp −θ (αx)β , x ≥ 0, θ+1

(1.1.61)

and

f (x; θ, α, β) =

i h i βθ2 h β β−1 α x + α2β x2β−1 exp −θ (αx)β , x > 0, θ+1

(1.1.62)

where θ, α, β are all positive parameters.

Remark 1.22. Ghitany et al. (2013) introduced the following distribution   β α F (x; α, β) = 1 − 1 + x exp [−βxα ] , x ≥ 0, β+1

which is quite similar to (1.1.61). The cd f F (x; α, β) has been characterized in Hamedani and Maadooliat (2017).

G. G. Hamedani

30

1.1.31

Marshall-Olkin Alpha Power (MOAP)

The cd f and pd f of MOAP are given, respectively, by  αG(x) −1 
, if α > 0, α 6= 0 G(x) F (x; α, θ) = (α−1)θ+(1−θ) α −1 , x ∈ R, G(x), if α = 0

(1.1.63)

and

f (x; α, θ) =

 

θ log(α)g(x)αG(x) 2

(α−1)[(α−1)θ+(1−θ)(αG(x) −1)]

g(x),

, if α > 0, α 6= 0 if α = 1

, x ∈ R,

(1.1.64)

where α, θ are positive parameters and G (x) is the baseline cd f with the corresponding pd f g (x). Remark 1.23. For α 6= 1, the cd f (1.1.63) can be written as F (x; α, θ) = G(x)

G1 (x) , 1 − (1 − θ) G1 (x)

x ∈ R,

where G1 (x) = α α−1−1 which is a cd f . The above distribution has been introduced before and characterized.

1.1.32

Zero Spiked Gamma Weibull (ZSGW)

The cd f and pd f of ZSGW are given, respectively, by F (x; φ, η) =

1 Γ (φ)

Z − log[1−G(x;η)] 0

vφ−1 e−v dv, x ∈ R,

(1.1.65)

and f (x; φ, η) =

1 g (x; η) {− log [1 − G (x; η)]}φ−1 , x ∈ R, Γ (φ)

(1.1.66)

where φ is a positive parameter and G (x; η) is the baseline cd f with the corresponding pd f g (x; η) which may depend on the parameter vector η. Remark 1.24. Cordeiro et al. (2016) proposed the following distribution " ( )# 1 Gα (x; τ) F (x; α, β, τ) = γ β, − log 1 − , x ∈ R, α Γ (β) Gα (x; τ) + G (x; τ) where α, β are positive parameters and G (x; τ) is the baseline cd f which may depend on the parameter τ . Taking α = 1 in F (x; α, β, τ), we arrive at (1.1.65). The cd f F (x; α, β, τ) has been characterized in Hamedani (2016).

Introduction

1.1.33

31

Inverted Nadarajah-Haghighi (INH)

The cd f and pd f of INH are given, respectively, by n
α o F (x; α, λ) = exp 1 − 1 + λx−1 , x ≥ 0,

(1.1.67)

and

f (x; α, λ) = αλx−2 1 + λx−1 where α, λ are positive parameters.


α−1

n
α o exp 1 − 1 + λx−1 , x > 0,

(1.1.68)

Remark 1.25. Asgharzadeh et al. (2016) introduced the following distribution n  α o F (x; α, θ, β) = exp 1 − 1 + θx−β , x ≥ 0,

where α, θ, β are all positive parameters. Clearly, F (x; α, θ, β) is more general than cd f (1.1.67) and has been characterized in Hamedani and Maadooliat (2017).

1.1.34

Marshall-Olkin Generated Gamma (MOGG)

The cd f and pd f of MOGG are given, respectively, by

F (x; α, λ) =

and

f (x; α, λ) =

i o n h    ;λ−2 α 1−ΓG λ−2 exp λ log(x)−µ σ    n h  i o  log(x)−µ  1−α¯ 1−ΓG λ−2 exp λ ;λ−2  σ     log(x)−µ  αΦ − σ   ¯ − log(x)−µ 1−αΦ  σ h   n i o    αΓG λ−2 exp λ log(x)−µ ;λ−2  σ  n h  i o  ¯ G λ−2 exp λ log(x)−µ 1−αΓ ;λ−2 σ

, if

λ>0

, if

λ=0,

, if

λ0

, if

λ=0,

, if

λ 0, σ > 0, λ ∈ R, µ ∈ R are parameters, ΓG (x; k) = Γ(k)

|λ| . Γ(λ−2 )

x ≥ 0,

(1.1.69)

x > 0, (1.1.70)

R x k−1 −w e dw and c (λ) = 0 w

32

G. G. Hamedani

Remark 1.26. Gui (2013) proposed the following distribution i p h  log(x)−µ α Φ − σ h  i p , x ≥ 0, F (x; α, µ, σ, p) = log(x)−µ 1−α Φ − σ

where α > 0, σ > 0, p > 0 and µ ∈ R are parameters. For p = 1, F (x; α, µ, σ, p) reduces to (1.1.69) for λ = 0. The cd f F (x; α, µ, σ, p) has been characterized in Hamedani (2019). Similar characterizations can be stated for the other two cases: λ < 0 and λ > 0.

1.1.35

Gamma Generalized Normal (GGN)

The cd f and pd f of GGN are given, respectively, by 1 F (x; α, µ, σ) = Γ (α)

Z − log[1−Φ( x−µ )] σ

t α−1 e−t dt, x ∈ R,

(1.1.71)


   α−1 φ x−µ x−µ σ f (x; α, µ, σ) = − log 1 − Φ , x ∈ R, Γ (α) σ

(1.1.72)

0

and

where α > 0, σ > 0, µ ∈ R are parameters and Φ (x) is the cd f of the standard normal with the corresponding pd f φ (x). Remark 1.27. Lima et al. (2015) proposed the distribution with the cd f (1.1.72), which has been characterized in Hamedani and Maadooliat (2017).

1.1.36

Transmuted Transmuted-G (TTG)

The cd f and pd f of TTG are given, respectively, by F (x; α, λ, ϕ) = (1 + α) H (x; λ, ϕ) − α [H (x; λ, ϕ)]2 , x ∈ R,

(1.1.73)

and f (x; α, λ, ϕ) = h (x; λ, ϕ) [1 + α − 2αH (x; λ, ϕ)] , x ∈ R,

(1.1.74)

where −1 ≤ α, λ ≤ 1 are parameters and H (x; λ, ϕ) is the cd f of the TG (Transmuted-G) distribution with the corresponding pd f h (x; λ, ϕ) which may depend on the parameter ϕ. Remark 1.28. Bourguignon et al. (2016) introduced the following distribution F (x; α, λ) = (1 + α) G (x; λ) − α [G (x; λ)]2 , x ∈ R,

which is the same as cd f (1.1.73) and has been characterized in Hamedani and Maadooliat (2017).

Introduction

1.1.37

33

Power Binomial Exponential 2 (PBE2)

The cd f and pd f of PBE2 are given, respectively, by   λθxα −λxα e , x ≥ 0, F (x; α, θ, λ) = 1 − 1 + 2−θ and   θ(λxα − 1) −λxα α−1 f (x; α, θ, λ) = αλx 1+ e , x > 0, 2−θ where α, λ > 0, 0 ≤ θ ≤ 1 are parameters. Remark 1.29. Taking θ =

1.1.38

2β 2β+1

(1.1.75)

(1.1.76)

in (1.1.75), we arrive at (1.1.61).

Beta Burr III (BBIII)

The cd f and pd f of BBIII are given, respectively, by 1 F (x; α, β, s, a, b) = B (a, b)

Z 1+( x )−α s 0

h

i−β

wa−1 (1 − w)b−1 dw, x ≥ 0,

(1.1.77)

and βa+1 (x/s)α α s (x/s)α+1 B (a, b) 1 + (x/s) ( β )b−1  (x/s)α , x > 0, × 1− 1 + (x/s)α 

αβ

f (x; α, β, s, a, b) =

(1.1.78)

where α, β, s, a, b are all positive parameters. h
−α i−β Remark 1.30. Taking Q (x) = 1 + xs in (1.1.77), we arrive at

Q(x) 1 wa−1 (1 − w)b−1 dw, x ≥ 0, B (a, b) 0 which has been introduced before and characterized in Hamedani (2016).

F (x; α, β, s, a, b) =

1.1.39

Z

Muth Generated (MG)

The cd f and pd f of MG are given, respectively, by 1

and

−α

F (x; α, β) = CG (x; β)−α e− α G(x;β) , x ∈ R,


−α 1 f (x; α, β) = Cg (x; β) 1 − αG (x; β)α G (x; β)−2α−1 e− α G(x;β) , x ∈ R,

where 0 < α ≤ 1, β > 0 are parameters and C = e1/α .

(1.1.79)

(1.1.80)

G. G. Hamedani

34

1.1.40

Weibull-Lindley (WLn)

The cd f and pd f of WLn are given, respectively, by θx −θx F (x; θ, α, β) = e−α{− log[1−(1+ θ+1 )e ]} , x ≥ 0,

(1.1.81)

and αβθ2 (1 + x) e−θx
×  θx e−θx (θ + 1) 1 − 1 + θ+1     β−1 θx − log 1 − 1 + e−θx θ+1 θx −θx × e−α{− log[1−(1+ θ+1 )e ]}, x > 0,

f (x; θ, α, β) =

(1.1.82)

x > 0, where θ, α, β are all positive parameters and. Remark 1.31. The cd f (1.1.81) is a special case of the cd f introduced by Tahir et al. (2016) (see (2.1.1) of the Leren et al). The distribution of Tahir et al. has been characterized in the same paper.

1.1.41

Weibull-G Power Series (WGPS)

The cd f and pd f of WGPS are given, respectively, by    h iβ  G(x;η) C θ 1 − exp −α G(x;η) F (x; α, β, θ, η) = , x ∈ R, C (θ)

(1.1.83)

and θαβg (x; η) (G (x; η))β−1
β+1 G (x; η)    h iβ  G(x;η) h iβ C0 θ 1 − exp −α G(x;η) −α G(x;η) × e G(x;η) , x ∈ R, C (θ)

f (x; α, β, θ, η) =

(1.1.84)

where α, β, θ, η are all positive parameters, G (x; η) is a baseline cd f with the correspondn 0 ing pd f g (x; η) and C (θ) = ∑∞ n=1 an θ is finite for an s positive real numbers.

1.1.42

Three Parameter Generalized Lindley (TPGL)

The cd f and pd f of TPGL are given, respectively, by α

(1 + λβ + λxα ) e−λx F (x; α, β, λ) = 1 − , x ≥ 0, 1 + λβ

(1.1.85)

Introduction

35

and α

αλ2 (1 + λβ + λxα ) e−λx , x > 0, f (x; α, β, λ) = 1 + λβ

(1.1.86)

where α, β, λ are all positive parameters. Remark 1.32. Ghitany et al. (2013) introduced the following distribution   1 + β + βxα F (x; α, β) = 1 − exp [−βxα ], x ≥ 0, 1+β

which is quite similar to (1.1.85). The cd f F (x; α, β) has been characterized in Hamedani and Maadooliat (2017).

1.1.43

Odd Lindley Exponentiated Weibull (OLEW)

The cd f and pd f of OLEW are given, respectively, by  n h iα o −xβ n h i o a + 1 − 1 − e −1 α β F (x; α, β, a) = 1 − 1 − 1 − e−x 1+a  

(1.1.87)

β h i o h i a2 αβxβ−1 e−x n β α −3 β α−1 1 − 1 − e−x 1 − e−x f (x; α, β, a) = 1+a  

x > 0,

(1.1.88)

, x ∈ R,

(1.1.89)

h i h i β α β α −a 1−e−x / 1− 1−e−x

×e

x ≥ 0,

,

and

h i h i β α β α − 1−e−x / 1− 1−e−x

×e

,

where α, β, a are all positive parameters.

1.1.44

Extended Odd Fréchet-G (EOF-G)

The cd f and pd f of EOF-G are given, respectively, by −

F (x; α, θ, ψ) = e

h

i 1−G(x;ψ)α θ G(x;ψ)α

and

f (x; α, θ, ψ) =

αθg (x; ψ) 1 − G (x; ψ)α G (x; ψ)αθ+1


θ−1

−

e

h

i 1−G(x;ψ)α β G(x;ψ)α

, x ∈ R,

(1.1.90)

where α > 0, θ > 0 are parameters and G (x; ψ) is a baseline cd f which may depend on the parameter vector ψ,with the corresponding pd f g (x; ψ).

G. G. Hamedani

36

Remark 1.33. Ahsan ul Haq and Elgarhy (2018) introduced the following distribution −

F (x; α, ψ) = e

h

i 1−K(x;ψ) θ K(x;ψ)

, x ∈ R.

where K (x; ψ) is a baseline cd f . Takin K (x; ψ) = G (x; ψ)α we arrive at the cd f (1.1.89). The cd f F (x; α, ψ) has been characterized in Hamedani (2017).

1.1.45

Alpha Power Transformation Poisson Lindley (APTPL)

The cd f and pd f of APTPL (for α 6= 1) are given, respectively, by F (x; α, θ, β) =

α1−e

−θxβ

[1+θxβ/(θ+1) ] − 1 α−1

and

, x ≥ 0,

(1.1.91)

β

(logα)βθ2 xβ−1 e−θx 1−e−θxβ [1+θxβ/(θ+1) ] α , f (x; α, θ, β) = (α − 1) (θ + 1)

(1.1.92)

x > 0, where α 6= 1 > 0, θ > 0, β > 0 are parameters.

Remark 1.34. The cd f (1.1.91) is a special case of the following distribution αG(x;ψ) − 1 , x ∈ R, α−1  β  where G (x; ψ) is a baseline cd f . Takin G (x; ψ) = 1 − e−θx 1 + θxβ/(θ+1) we arrive at the cd f (1.1.91), which has been characterized in Hamedani (2017). F (x; α, ψ) =

1.1.46

Alpha Logarithm Transmuted Fréchet (ALTF)

The cd f and pd f of ALTF (for α 6= 1) are given, respectively, by h i β log α − (α − 1) e−(λ/x) F (x; α, β, λ) = 1 − , x ≥ 0, log(α)

(1.1.93)

and β

(α − 1) λβ βx−(β+1) e−(λ/x) h i , x > 0, f (x; α, β, λ) = β log(α) α − (α − 1) e−(λ/x)

(1.1.94)

where α > 0, α 6= 1, β > 0, λ > 0 are parameters.

Remark 1.35. Hakamipour et al. (2012) proposed the following distribution F (x) = F (x; α, p) = 1 −

 1 log p + (1 − p) [G1 (x)]α , x ≥ 0, log p

where α > 0, p ∈ (0, 1) are parameters and G1 (x) is a baseline cd f . It is clear that the β condition p ∈ (0, 1) can be replaced by p > 0, p 6= 1. Then, takin [G1 (x)]α = e−(λ/x) , x ≥ 0,

Introduction

37

we arrive at the cd f (1.1.93). The distribution proposed by Hakamipour et al. (2012) has been characterized in Hamedani (2016).

1.1.47

Burr-Weibull Power Series (BWPS)

The cd f and pd f of BWPS are given, respectively, by   β C θ (1 + xc )−k e−αx , x ≥ 0, F (x; α, β, θ, c, k) = 1 − C (θ)

(1.1.95)

and d F (x; α, β, θ, c, k), x > 0, (1.1.96) dx n where α, β, θ, c, k are all positive parameters and C (θ) = ∑∞ n=1 an θ is finite and {an }n≥1 is a sequence of positive real numbers. f (x; α, β, θ, c, k) =

Remark 1.36. Condino and Domma (2017) proposed the following distribution F (x; α, θ, ν) = 1 −

C (θ [1 − G (x)]) , x ∈ R, C (θ)

where α, θ, ν are all positive parameters and G (x) is a baseline cd f . Taking G (x) = β 1 − (1 + xc )−k e−αx , x ≥ 0, we arrive at the cd f (1.1.95). The cd f F (x; α, θ, ν) has been characterized in Hamedani (2017).

1.1.48

Zografos-Balakrishnan Fréchet (ZBFr)

The cd f and pd f of ZBFr are given, respectively, by 1 F (x; θ, ζ, λ) = Γ (λ)

Z − log 1−e−ζθ x−θ h

0

i

t λ−1 e−t dt, x ≥ 0,

(1.1.97)

and θζθ x−(θ+1)e f (x; θ, ζ, λ) = Γ (λ)

−ζθ x−θ

 h iλ−1 θ − log 1 − e−ζ x−θ ,

(1.1.98)

x > 0, where θ, ζ, λ are all positive parameters.

Remark 1.37. Cordeiro et al. (2016) proposed the following distribution βα F (x; α, β, τ) = Γ (α)

Z

0

G(x;τ) G(x;τ)

t α−1 e−βt dt, x ∈ R,

where G (x; τ) is a baseline cd f . Taking G (x; τ) =

h i θ log 1−e−ζ x−θ h i, − 1−log 1−e−ζθ x−θ

x ≥ 0, we arrive at

the cd f (1.1.97). The cd f F (x; α, β, τ) has been characterized in Hamedani (2018b).

G. G. Hamedani

38

1.1.49

Cubic Transmuted Weibull (CTW)

The cd f and pd f of CTW are given, respectively, by    β β F (x; α, β, λ) = 1 − e−αx 1 + λe−2αx , x ≥ 0,

(1.1.99)

and

h i β β β f (x; θ, ζ, λ) = αβxβ−1 e−αx 1 − 2λe−αx + 3λe−2αx ,

(1.1.100)

x > 0, where α, β positive and λ (|λ| ≤ 1) are parameters.

1.1.50

Cubic Rank Transmuted Kumaraswamy (CRTKw)

The cd f and pd f of CRTKw are given, respectively, by h i h i2 F (x; a, b, λ1, λ2 ) = λ1 1 − (1 − xa )b + (λ2 − λ1 ) 1 − (1 − xa )b h i3 + (1 − λ2 ) 1 − (1 − xa )b , 0 ≤ x ≤ 1,

and

(1.1.101)

f (x; a, b, λ1, λ2 ) = abxa−1 (1 − xa )b−1 h i     λ1 + 2 (λ2 − λ1 ) 1 − (1 − xa )b h i2 , 0 < x < 1,  +3 (1 − λ ) λ + 2 (λ − λ ) 1 − (1 − xa )b  2 1 2 1

(1.1.102)

0 < x < 1, where a > 0, b > 0, λ1 ∈ [0, 1] and λ2 ∈ [−1, 1] are parameters.

1.1.51

Cubic Transmuted Weibull (CTW)

The cd f and pd f of CTW are given, respectively, by F (x; k, λ, λ1 , λ2 ) = 1 + (λ1 + λ2 − 1) e−(x/λ) k

k k

− (λ1 + 2λ2 )e−2(x/λ) + λ2 e−3(x/λ) ,

(1.1.103)

x ≥ 0, and k

f (x; k, λ, λ1, λ2 ) = kλ−k xk−1 e−3(x/λ) × h i k k (1 − λ1 − λ2 ) e2(x/λ) + 2 (λ1 + 2λ2 ) e(x/λ) − 3λ2 ,

x > 0, where k > 0, λ > 0, λ1 ∈ [−1, 1] and λ2 ∈ [−1, 1] are parameters.

(1.1.104)

Introduction

1.1.52

39

Type II Kumaraswamy Half Logistic-Generated (TIIKwHL-G)

The cd f and pd f of TIIKwHL-G are given, respectively, by a b   1 − G (x) , x ∈ R, F (x; a, b) = 1 − 1 + G (x)

and

f (x; a, b) =

2abg (x) [1 + G (x)]2



1 − G (x) 1 + G (x)

a−1    b−1 1 − G (x) a 1− , 1 + G (x)

(1.1.105)

(1.1.106)

x ∈ R, where a > 0, b > 0 are parameters and G (x) is a baseline cd f with the corresponding pd f g (x). Remark 1.38. Hadique et al. (2019) proposed the following distribution F (x; α, a, b, η) =

(



αG(x; η) 1− 1 − αG(x; η)

a )b

, x ∈ R,

where G (x; η) is a baseline cd f . Taking G (x; η) = G (x) , x ∈ R and α = 1/2, we arrive at the cd f (1.1.105). The cd f F (x; α, β, η) has been characterized in Hamedani (2019).

1.1.53

Odd Log-Logistic Generalized Inverse Gaussian (OLLGIG)

The cd f and pd f of OLLGIG are given, respectively, by F (x; µ, σ, ν, τ) =

Gµ,σ,ν (x)τ , x ≥ 0, Gµ,σ,ν (x)τ + Gµ,σ,ν (x)τ

(1.1.107)

and  ν    b τxν−1 1 bx µ f (x; µ, σ, ν, τ) = exp − 2 + µ 2Kν (σ−2 ) 2σ µ bx  −2 × {η (x) [1 − η (x)]}τ−1 η (x)ν + [1 − η (x)]ν ,

(1.1.108)

x > 0,

where µ, σ, τ all hpositive, νi∈ R are parameters and  R x b ν τt ν−1 µ 1 bt Gµ,σ,ν (x) = exp − 2σ2 µ + bt dt and Kν (t) = 0 µ 2Kν (σ−2 ) 
 1 R ∞ ν−1 −1 exp − (1/2)t u + u du is the modified Bessel function of the third 2 0 x kind. Remark 1.39. Alizadeh et al. (2017) proposed the following distribution F (x) =

G (x; γ)α , x ∈ R, G (x; γ)α + βG (x; γ)α

where G (x; γ) is a baseline cd f . The characterizations of the cd f (1.1.107) are similar to those of F (x) which have appeared in Hamedani (2019).

G. G. Hamedani

40

1.1.54

General Class (GC)

The cd f and pd f of GC are given, respectively, by F (x; τ) =

1 {F1 (x; τ) + κF2 (x; τ)} , x ∈ R, 1 + κτ

(1.1.109)

and f (x; τ) =



 1 + τh (x) g (x) , x ∈ R, 1 + κτ

(1.1.110)

where τ ≥ 0 is a parameter, g is a pd f , h is a positive continuous function such that Eg [h (Y )] = κ < ∞, Y ∼ g and F1 (x; τ) , F2 (x; τ) are cd f s of X1 ∼ g (x) and X2 ∼ (1/κ)h (x) g (x) , respectively.

1.1.55

Poisson Burr Type X Log-Logistic (PBXLL)

The cd f and pd f of PBXLL are given, respectively, by   2α υ −λ 1−e−x

and

1−e F (x; λ, υ, α) = 1 − e−λ

, x ≥ 0,

υ−1 2αλυx2α−1  −λ −x2α f (x; λ, υ, α) = 1 − e e −λ 1−e



 2α υ

1−e−x

(1.1.111)

,

(1.1.112)

x > 0, where λ, υ, α are all positive parameters.

Remark 1.40. Nadarajah et al. (2009) proposed the following distribution 1 − e−λG(x) , x ∈ R, 1 − e−λ   2α υ where G (x) is a baseline cd f . Taking G (x) = 1 − e−x , x ≥ 0, in F (x), we arrive at cd f (1.1.111). The characterizations of the cd f (111) are similar to those of F (x) which have appeared in Hamedani (2016). F (x) =

1.1.56

New Odd Generalized Exponential-Exponential (NOGE-E)

The cd f and pd f of NOGE-E are given, respectively, by

and

 β λx F (x; α, β, λ) = 1 − e−α(e −1) , x ≥ 0, f (x; α, β, λ) = αβλeλx e−α(e

λx −1

 β−1 ) 1 − e−α(eλx−1) ,

x > 0, where α, β, λ are all positive parameters.

(1.1.113)

(1.1.114)

Introduction

41

Remark 1.41. Cordeiro et al. (2016) proposed the following distribution  β F (x; α, β) = 1 − {1 − G (x)}α , x ∈ R,

λx where G (x) is a baseline cd f . Taking G (x) = 1 − e−α(e −1), x ≥ 0 and α = 1 in F (x; α, β), we arrive at cd f (1.1.113). The characterizations of (1.1.113) are similar to those of the cd f F (x; α, β) which have appeared in Hamedani (2016).

1.1.57

Kumaraswamy Extension Exponential (KEE)

The cd f and pd f of KEE are given, respectively, by n  o α a b F (x; a, b, α, λ) = 1 − 1 − 1 − e1−(1+λx) , x ≥ 0,

and

α

f (x; a, b, α, λ) = abαλ (1 + λx)α−1 e1−(1+λx) n  o α a b−1 1 − 1 − e1−(1+λx) ,

  α a−1 1 − e1−(1+λx) ×

(1.1.115)

(1.1.116)

x > 0, where a, b, α, λ are all positive parameters.

Remark 1.42. Al-babtain et al. (2018) proposed the following distribution F (x; α, β, θ, γ, a, b,λ)  h iαa h  α ia b −(θx+γxβ ) −(θx+γxβ ) , x ≥ 0. = 1− 1− 1−e 1+λ−λ 1−e

where G (x) is a baseline cd f . Taking λ = 0 in F (x; α, β, θ, γ, a, b, λ), we arrive at a more general version of cd f (1.1.115). The cd f F (x; α, β, θ, γ, a, b, λ) has been characterized in Hamedani (2018a).

1.1.58

Extended Enlarg Transmuted Exponential (EETE)

The cd f and pd f of EETE are given, respectively, by  α −λ F (x; α, β, λ) = 1 − e−β(1−e )x , x ≥ 0,

(1.1.117)

and

   α−1 −λ −λ f (x; α, β, λ) = αβ 1 − e−λ e−β(1−e )x 1 − e−β(1−e )x , x > 0,

(1.1.118)

where α, β, λ are all positive parameters.
Remark 1.43. Taking γ = β 1 − e−λ , the cd f (1.1.117) is simply the α power of the exponential distribution.

G. G. Hamedani

42

1.1.59

Marshall-Olkin Extended Power Function (MOEPF)

The cd f and pd f of MOEPF are given, respectively, by   α  γ 1 − ψx   α  , 0 ≤ x ≤ ψ, F (x; α, γ, ψ) = 1 −  α x x + γ 1 − ψ ψ

(1.1.119)

and

−α α−1

f (x; α, γ, ψ) = αγψ

x

 α   α −2 x x +γ 1− , x > 0, ψ ψ

(1.1.120)

where α, γ, ψ are all positive parameters.  α Remark 1.44. Taking G (x) = ψx , 0 ≤ x ≤ ψ, the cd f (1.1.119) can be written as F (x; α, γ, ψ) =

G (x) , 0 ≤ x ≤ ψ. G (x) + γG (x)

Cordeiro et al. (2016) introduced the following distribution F (x; α, θ, τ) =

(G (x; τ))αθ h iα , x ∈ R, (G (x; τ))αθ + 1 − (G (x; τ))θ

which is more general than the cd f F (x; α, γ, ψ) = in Hamedani (2019).

1.1.60

G(x) , which G(x)+γG(x)

has been characterized

Generalized Odd Log-Logistic Exponential (GOLLE)

The cd f and pd f of GOLLE are given, respectively, by

and


αθ 1 − e−λx F (x; α, θ, λ) =
αθ h
θ iα , x ≥ 0, 1 − e−λx + 1 − 1 − e−λx
αθ−1 h
θ iα−1 αθλe−λx 1 − e−λx 1 − 1 − e−λx , x > 0, f (x; α, θ, λ) = n
αθ h
θ iα o2 −λx −λx 1−e + 1− 1−e

(1.1.121)

(1.1.122)

where α, θ, λ are all positive parameters. Remark 1.45. Taking G (x) = 1 − e−λx F (x; α, θ, λ) =


θ

, x ≥ 0, the cd f (1.1.121) can be written as

G (x)α , x ≥ 0. G (x)α + G (x)α

Introduction

43

Cordeiro et al. (2016) introduced the following distribution F (x; α, θ, τ) =

(G (x; τ))αθ h iα , x ∈ R, (G (x; τ))αθ + 1 − (G (x; τ))θ

is more general than the cd f F (x; α, θ, λ) = terized in Hamedani (2019).

1.1.61

G(x)α , G(x)α +G(x)α

x ≥ 0, which has been charac-

Exponentiated Transmuted Power Function (ETPF)

The cd f and pd f of ETPF are given, respectively, by

and

 αθ   α θ x x F (x; α, β, θ, γ) = 1+γ−γ , 0 ≤ x ≤ β, β β

 αθ−1   α θ x αθ x 1 + γ − 2γ f (x; α, β, θ, γ) = β β β   α θ−1 x × 1+γ−γ , 0 < x < β, β

(1.1.123)

(1.1.124)

where α, β, θ, γ are all positive parameters.

1.1.62

Type II Half Logistic Weibull (TIIHLW)

The cd f and pd f of TIIHLW are given, respectively, by

and

 γ λ 2 1 − e−δx F (x; λ, δ, γ) =   , x ≥ 0, γ λ 1 + 1 − e−δx γ γ λ−1 2λδγxγ−1 e−δx 1 − e−δx f (x; λ, δ, γ) = , n   o2 γ λ −δx 1+ 1−e

(1.1.125)

x > 0,

(1.1.126)

where λ, δ, γ are all positive parameters.

Remarks 1.46. (a) Hassan et al. (2017) proposed the following distribution F (x; λ, ζ) = γ

2G (x; ζ)λ 1 + G (x; ζ)λ

, x ∈ R.

Taking G (x; ζ) = 1 − e−δx , x ≥ 0, we arrive at the cd f (1.1.125). The characterizations of (1.1.125) are similar to those of F (x; λ, ζ) which have appeared in Hamedani (2018a). (b) Elgarhy et al. (2018) considered the following distribution

44

G. G. Hamedani  λ 2 1 − e−δx F (x; λ, δ) =  λ , x ≥ 0, 1 + 1 − e−δx

which is clearly a special case of (1.1.125). (c) Ahsan ul Haq et al. (2018) considered the following distribution h i 2 λ 2 1 − e−δx F (x; λ, δ) =   , x ≥ 0, 2 λ 1 + 1 − e−δx

which is a slight improvement on Elgarhy et al. (2018), but still an special case of (1.1.125). (d) It should be mention that the distribution in (b) was proposed prior to that of (c) and of course, prior to that of (1.1.125).

1.1.63

Exponentiated Generalized Inverse Rayleigh (EGIR)

The cd f and pd f of EGIR are given, respectively, by

and

h   i −2 α γ F (x; α, λ, γ) = 1 − 1 − e−(λx) , x ≥ 0,   −2 −2 α−1 2αγ f (x; α, λ, γ) = 2 3 e−(λx) 1 − e−(λx) × λx h   i −2 α γ−1 1 − 1 − e−(λx) , x > 0,

(1.1.127)

(1.1.128)

where α, λ, γ are all positive parameters.

Remark 1.47. The cd f (1.1.127) is a special case of the one given in Remark 1.13.

1.1.64

Generalized Transmuted Gompertz-Makeham (GTGM)

The cd f and pd f of GTGM are given, respectively, by  

α δ(1−eβx)−γδx F (x; α, β, δ, γ, ε, λ) = 1 − e β ×     α βx β ε(1−e )−γεx 1 − λ + λe , x ≥ 0,

(1.1.129)

and   α α(1−eβx)−γδx βx f (x; α, β, δ, γ, ε, λ) = γ + αe e β ×     α βx β ε(1−e )−γεx δ(1 − λ) + λ (δ + ε) e , x > 0, where α, β, δ, γ, ε all positive and λ (|λ| ≤ 1) are parameters.

(1.1.130)

Introduction

1.1.65

45

Jamal Weibull-X (JW-X)

The cd f and pd f of JW-X are given, respectively, by − log(1 − G (x; γ)a ) F (x; α, β, γ, a) = 1 − exp −α 1 − G (x; γ)a "



β #

, x ∈ R,

(1.1.131)

and

f (x; α, β, γ, a) =

αβag (x; γ)G (x; γ)a−1 {log[1 − G (x; γ)a ]}β [1 − G (x; γ)a ]

β+1

− log(1 − G (x; γ)a ) exp −α 1 − G (x; γ)a "



β #

×

,

(1.1.132)

x ∈ R, where α, β, γ, a are all positive parameters and G (x; γ) is the baseline cd f with the corresponding pd f (x; γ).

1.1.66

Nasir Logistic-X (NL-X)

The cd f and pd f of NL-X are given, respectively, by  −1    log[− log(1 − G (x; γ)a )] , x ∈ R, F (x; λ, γ, a) = 1 + exp −λ 1 − G (x; γ)a

(1.1.133)

and d F (x; λ, γ, a), x ∈ R, dx where λ, γ, a are all positive parameters and G (x; γ) is the baseline cd f . f (x; λ, γ, a) =

(1.1.134)

Remark 1.48. The characterizations of cd f (1.1.133) are similar to those of cd f (1.1.59).

1.1.67

Jamal Logistics-X (JL-X)

The cd f and pd f of JL-X are given, respectively, by     −1 G (x; γ)a F (x; λ, γ, a) = 1 + exp −λ log , x ∈ R, 1 − G (x; γ)a

(1.1.135)

and d F (x; λ, γ, a), x ∈ R, dx where λ, γ, a are all positive parameters and G (x; γ) is the baseline cd f . f (x; λ, γ, a) =

(1.1.136)

G. G. Hamedani

46

1.1.68

Nasir Weibull-Generalized (NW-G)

The cd f and pd f of NW-G are given, respectively, by " β #  − log [G (x; γ)] , x ∈ R, F (x; α, β, γ) = exp −α G (x; γ)

(1.1.137)

and d F (x; α, β, γ), x ∈ R, dx where α, β, γ are all positive parameters and G (x; γ) is the baseline cd f . f (x; λ, γ, a) =

1.1.69

(1.1.138)

Weibull Pareto (WP)

The cd f and pd f of WP are given, respectively, by −

F (x; α, θ, b) = 1 − e

h

( θx )

α

−1

ib

, x ≥ θ,

(1.1.139)

and ib−1 h x α ib bαxα−1 h x α − ( ) −1 f (x; α, θ, b) = , x > θ, −1 e θ θα θ which has been characterized in Hamedani (2019).

1.1.70

(1.1.140)

Slashed Power-Lindley (SPL)

The cd f and pd f of SPL are given, respectively, by F (x; α, θ, b) =

Z x 0

f (u; α, θ, b)du, x ≥ 0,

(1.1.141)

and

f (x; α, θ, b) =

bx−(b+1)

b × (θ + 1) θ α      b b b α +1 b+α −θxα α +θ+1 γ + 1, θx − θ x e α α

x > 0, where α, θ, b are all positive parameters and γ (α, x) =

R x α−1 −u e du. 0 u

(1.1.142)

Introduction

1.1.71

47

Exponentiated Generalized Pareto (EGP)

The cd f and pd f of EGP are given, respectively, by ex F (x; α, d) = 1 − 1 + d 

and f (x; α, d) =

−α

d F (x; α, d), dx

, x ∈ R,

(1.1.143)

x ∈ R,

(1.1.144)

where α and d are positive parameters. Remark 1.49. Taking G (x) = (1.1.143).

1.1.72

ex ex +2d ,

x ∈ R, in the cd f (1.1.105) we arrive at the cd f

Inverse Weighted Lindley (IWL)

The cd f and pd f of IWL are given, respectively, by

φ −1 Γ φ, λx−1 (λ + φ) + λx−1 e−λx , x ≥ 0, F (x; φ, λ) = Γ (φ)(λ + φ)

(1.1.145)

and f (x; φ, λ) =


−1 λφ+1 x−φ−1 1 + x−1 e−λx , Γ (φ)(λ + φ)

where φ, λ are positive parameters and Γ (x, y) =

1.1.73

x > 0,

(1.1.146)

R ∞ y−1 −w e dw. x w

Unit-Inverse Gaussian (UIG)

The cd f and pd f of UIG are given, respectively, by F (x; µ, λ) =

Z x 0

f (u; µ, λ)du, 0 ≤ x ≤ 1,

(1.1.147)

and

f (x; φ, λ) =

r

λ 1 × 2π x (− logx)3/2

 λ 2 (logx + µ) , 0 < x < 1, 2µ2 log x  ∞ i f 0 < λ2 < 1, 2µ where µ, λ are positive parameters and f (0) = , f (1) = 0. λ exp



0 if

2µ2

≥ 1,

(1.1.148)

48

1.1.74

G. G. Hamedani

Weibull-Moment Exponential (WME)

The cd f and pd f of WME are given, respectively, by     x b  −    1 − 1 + βx e β   x  , x ≥ 0, F (x; a, b, β) = 1 − exp −a   −     1 + βx e β

(1.1.149)

and

 x ib−1 h  −β x 1 − 1 + β e abxe f (x; a, b, β) =  x ib+1 × h β2 − x 1+ β e β    x b   −  x   1− 1+ β e β     x  , exp −a     1 + βx e− β − βx

(1.1.150)

x > 0, where a, b, β are all positive parameters.

Remark 1.50. Bourguignon et al. (2014) proposed the following distribution (  ) K (x; γ) b F (x; a, b, β) = 1 − exp −a , x ∈ R, 1 − K (x; γ)   x − where K (x; γ) is the baseline cd f . Taking K (x; γ) = 1 − 1 + βx e β , x ≥ 0 in F (x; a, b, β), we arrive at the cd f (1.1.149). The cd f F (x; a, b, β) has been characterized in Hamedani (2016).

1.1.75

Generalized Odd Burr III-G (GOBIII-G)

The cd f and pd f of GOBIII-G are given, respectively, by 1 − G (x; ζ)α F (x; c, k, α, ζ) = 1 + G (x; ζ)α and





f (x; c, k, α, ζ) = ckαg (x; ζ)



c −k

1 − G (x; ζ)α

, x ∈ R,

c−1

[G (x; ζ)]cα+1   c −k−1 1 − G (x; ζ)α 1+ , G (x; ζ)α

(1.1.151)

× (1.1.152)

x ∈ R, where c, k, α, ζ are all positive parameters and G (x; ζ) is a baseline cd f with the corresponding pd f g (x; ζ).

Introduction

49

Remark 1.51. Jamal et al. (2017) introduced the following distribution    −k 1 − K (x; θ) c , x ∈ R, F (x; c, θ, k) = 1 + K (x; θ)

where K (x; θ) is a baseline cd f . Taking K (x; γ) = G (x; ζ)α , x ∈ R, in F (x; c, θ, k), we arrive at the cd f (1.1.151). The cd f F (x; c, θ, k) has been characterized in Hamedani (2018b).

1.1.76

Generalized Odd Fréchet-G (GOFr-G)

The cd f and pd f of GOFr-G are given, respectively, by (  θ ) 1 − G (x; ψ)α F (x; α, θ, ψ) = exp − , x ∈ R, G (x; ψ)α

(1.1.153)

and

f (x; α, θ, ψ) = αθg (x; ψ)



1 − G (x; ψ)α

θ−1

[G (x; ψ)]αθ+1 (  θ ) 1 − G (x; ψ)α exp − , G (x; ψ)α

× (1.1.154)

x ∈ R, where α, θ, ψ are all positive parameters and G (x; ψ) is a baseline cd f with the corresponding pd f g (x; ψ). Remark 1.52. Alizadeh et al. (2016) proposed the following distribution (  θ ) K (x; γ) F (x; α, γ) = exp − , x ∈ R, K (x; γ) where K (x; θ) is a baseline cd f . Taking K (x; γ) = G (x; ζ)α , x ∈ R, in F (x; α, γ), we arrive at the cd f (1.1.153). The cd f F (x; α, γ) has been characterized in Hamedani and Maadooliat (2017).

1.1.77

Type I Half Logistic Power Lindley (TIHLPL)

The cd f and pd f of TIHLPL are given, respectively, by

F (x; α, θ, λ) = and

1 − e−θλx

α

h

α

θx 1 + θ+1

iλ

  , x ≥ 0, α θxα λ 1 + e−θλx 1 + θ+1

(1.1.155)

G. G. Hamedani

50

iλ−1 h
α θxα 2λαθ2 xα−1 + x2α−1 e−θλx 1 + θ+1 , f (x; α, θ, λ) = n  o2  α θxα λ (θ + 1) 1 + e−θλx 1 + θ+1

(1.1.156)

x > 0, where α, θ, λ are all positive parameters.

Remark 1.53. Dias et al. (2016) introduced the following distribution F (x; α, λ, p) =

(


λ ) α 1 − G (x) ,
λ 1 − p G(x)

x ∈ R,

where α > h 0, λ > i0 , p < 1 are parameters and G (x) is a baseline cd f . Taking G (x) = θxα −θxα 1−e 1 + θ+1 , x ≥ 0 and p = −1 in F (x; α, λ, p), we arrive at the cd f (1.1.155). The cd f F (x; α, λ, p) has been characterized in Hamedani and Safavimanesh (2017).

1.1.78

New Alpha-Power Transformation (NAPT)

The cd f and pd f of NAPT are given, respectively, by ( G(x;η)αG(x;η) , α

F (x; α, η) =

G(x;η)

α 6= 1 , α=1

, x ∈ R,

(1.1.157)

and g (x; η) αG(x;η) [1 + log(α)G (x; η)] , α 6= 1, α x ∈ R, where α, η are positive parameters. f (x; α, η) =

(1.1.158)

Remark 1.54. Ahmad et al. (2018) introduced a distribution similar to that of (1.1.157), called TIAPT. The characterizations of NAPT are similar to those of TIAPT which have been reported in Ahmad et al. (2018).

1.1.79

Functional Weighted Exponential (FWE)

The cd f and pd f of FWE are given, respectively, by F (x; α, σ, θ) =

Z x 0

f (u; α, σ, θ)du, x ≥ 0,

(1.1.159)

and f (x; α, σ, θ) =

θe−θx αθx2 e−θx/σ + , x > 0, 1 + αx2 σ(σ2 + αx2 )

where α, σ, θ are all positive parameters.

(1.1.160)

Introduction

1.1.80

51

Odd Burr III G-Negative Binomial (OBIIIGNB)

The cd f and pd f of OBIIIGNB are given, respectively, by F (x; c, k, η, β, s) = x ≥ 0, and

(1 − β)−s − {1 − β [1 − Bc,k,η (x)]}−s , (1 − β)−s − 1

(1.1.161)

sβbc,k,η (x) {1 − β [1 − Bc,k,η (x)]}−s−1 , (1.1.162) (1 − β)−s − 1 n  c o−k x > 0, where c, k, η, β, s are all positive parameters, Bc,k,η (x) = 1 + 1−R(x;η) with R(x;η) f (x; c, k, η, β, s) =

bc,k,η (x) = B0c,k,η (x) and R (x; η) is the baseline cd f .

1.1.81

Type I Half-Logistic Exponential (TIHLE)

The cd f and pd f of TIHLE are given, respectively, by F (x; α, λ) =

1 − e−αλx , x ≥ 0, 1 + e−αλx

(1.1.163)

and 2αλe−αλx f (x; α, λ) =  2 , x > 0, 1 + e−αλx

(1.1.164)

where α, λ are positive parameters.

Remark 1.55. The cd f (1.1.163) is a special case of the cd f (1.1.155).

1.1.82

Unit-Marshall-Olkin Extended Exponential (UMOEE)

The cd f and pd f of UMOEE are given, respectively, by F (x; α, λ) = and f (x; α, λ) = 

where α, λ are positive parameters.

αxλ , 0 ≤ x ≤ 1, 1 − (1 − α) xλ αλxλ−1 1 − (1 − α) xλ

2 , 0 < x < 1,

(1.1.165)

(1.1.166)

Remark 1.56. The cd f (1.1.165) is a special case of the cd f given in Remark 1.23 with G (x) = xλ , x ∈ [0, 1].

G. G. Hamedani

52

1.1.83

Generalization of Two-Parameter Lindley (GTPL)

The cd f and pd f of GTPL are given, respectively, by   α θxα e−θx , x ≥ 0, F (x; α, β, θ) = 1 − 1 + βθ + 1 and αθ2 α−1 α x (β + xα ) e−θx , x > 0, βθ + 1 where α > 0, θ > 0 and β (βθ > −1) are parameters. f (x; α, β, θ) =

(1.1.167)

(1.1.168)

Remark 1.57. The cd f (1.1.167) is essentially similar to that of (1.1.61).

1.1.84

Odd Lindley Fréchet (OLiFr)

The cd f and pd f of OLiFr are given, respectively, by 

and

β

−( αx )



  θ+ 1−e  −θe−( αx )β    exp , x ≥ 0, F (x; α, β, θ) = 1 − β β  −( αx )  −( αx ) 1 − e (1 + θ) 1 − e   α β  −θe−( αx )β  βαβ θ2 x−β−1 e−( x ) , x > 0, f (x; α, β, θ) =  3 exp β β  −( αx )  −( αx ) 1 − e (1 + θ) 1 − e

(1.1.169)

(1.1.170)

where α, β and θ are all positive parameters.

Remark 1.58. The cd f (1.1.169) is a special case of cd f (1.1.87).

1.1.85

Topp-Leone Mukherjee-Islam (TLMI)

The cd f and pd f of TLMI are given, respectively, by      α x p x 2p F (x; α, θ, p) = 2 − , 0 ≤ x ≤ θ, θ θ and

(1.1.171)

 x  p iα h  x  p iα−1 2αp αp−1 h x 1 − 2 − , 0 < x < θ, (1.1.172) θαp θ θ where α, θ and p are all positive parameters.
p Remark 1.59. Taking G (x) = θx , 0 ≤ x ≤ θ, in cd f (1.1.73) reduces to the cd f (1.1.171). f (x; α, θ, p) =

Introduction

1.1.86

53

Weibull-Lomax (WL)

The cd f and pd f of WL are given, respectively, by F (x; β, λ, c) = 1 − exp {− [β log (1 + λx)]c } , x ≥ 0,

(1.1.173)

and βcλ [β log(1 + λx)]c−1 exp {− [β log(1 + λx)]c }, x > 0, f (x; β, λ, c) = (1 + λx)

(1.1.174)

where β, λ, c are all positive parameters. Remark 1.60. Ghosh and Nadarajah (2018), proposed the following distribution   1 c F (x; γ, c) = 1 − exp − c [− log(1 − G (x))] , x ∈ R, γ where G (x) is a baseline cd f . Taking G (x) = 1 − (1 + λx)−1 , x ≥ 0 in F (x; γ, c), we arrive at the cd f (1.1.173). Thus, WL is a special case of F (x; γ, c).

1.1.87

Zero Truncated Poisson Topp-Leone Exponentiated Weibull (ZTPTLEW)

The cd f and pd f of ZTPTLEW are given, respectively, by

F (x; α, λ, a, b) =

n h i o  h i α  b a b a −x −x 1 − exp −λ 1 − e − 1−e +2 1 − e−λ

x ≥ 0, and

,

(1.1.175)

d F (x; α, λ, a, b), x > 0, (1.1.176) dx where α, λ, a, b are all positive parameters. nh ia o  h ia α  −xb −xb Remark 1.61. Taking G (x) = exp 1−e − 1−e +2 , x ≥ 0 in f (x; α, λ, a, b) =

Remark 1.39, we arrive at the cd f (1.1.175).

1.1.88

Zero Truncated Poisson Topp-Leone Burr XII (ZTPTLBXII)

The cd f and pd f of ZTPTLBXII are given, respectively, by  h ib  −2β α 1 − exp −a 1 − (1 + x ) F (x; α, β, a, b) = , 1 − e−a x ≥ 0, and

(1.1.177)

G. G. Hamedani

54

h ib−1 2abαβ α−1 α −2β−1 α −2β x (1 + x ) 1 − (1 + x ) × 1− e−a  h ib exp −a 1 − (1 + xα )−2β , (1.1.178)

f (x; α, β, a, b) =

x > 0, where α, β, a, b are all positive parameters.

1.1.89

General Transmuted Family (GTF)

The cd f and pd f of GTF are given, respectively, by "

k

F (x; λi , i = 0, 1, ..., k) = G (x) 1 + ∑ λi G (x)

x ∈ R, and k

i

i=1



i

#

,

  i f (x; λi , i = 0, 1, ..., k) = g (x) ∑ ∑ (−1) (1 + j) λi [G (x)] j , j i=0 j=0 j

(1.1.179)

(1.1.180)

x ∈ R, where λ0 = 1, λ1 ∈ [−1, 1] and λi ∈ [0, 1], i = 2, 3, ..., k are parameters and G (x) is a baseline cd f with corresponding pd f g (x).

1.1.90

Poisson Topp-Leone Inverse Weibull (PTLIW)

The cd f and pd f of PTLIW are given, respectively, by   θ  −θbδβ x−β −bδβ x−β 1 − exp −ae 2−e F (x; β, θ, δ, a, b) = , 1 − e−a x ≥ 0, and β x−β

2θβabδβ x−(β+1)e−θbδ

(1.1.181)

  β −β θ−1 2 − e−bδ x

× f (x; β, θ, δ, a, b) = 1 − e−a  n o  θ  −θbδβ x−β −θbδβ x−β −bδβ x−β 1−e exp −ae 2−e ,

(1.1.182)

x > 0, where β, θ, δ, a, b are all positive parameters.

1.1.91

Odd Burr-G Poisson (OBGP)

The cd f and pd f of OBGP are given, respectively, by F (x; λ, η, c, k) =

1 − exp {−λBc,k (x)} , x ∈ R, 1 − e−λ

(1.1.183)

Introduction

55

and λbc,k (x) exp {−λBc,k (x)} , x ∈ R, (1.1.184) 1 − e−a n  c o−k G(x;η) where λ, c, k are all positive parameters and Bc,k (x) = 1 − 1 + 1−G(x;η) , x ∈ R with corresponding pd f bc,k (x) and with G (x; η) as a baseline cd f . f (x; λ, η, c, k) =

Remark 1.62. Nadarajah et al. (2009) proposed the following distribution F (x; λ) =

1 − e−λG1 (x) , x ∈ R. 1 − e−λ

Taking G1 (x) = Bc,k (x), we arrive at the cd f (1.1.183). The cd f F (x; λ) was characterized in Hamedani (2016).

1.1.92

Exponentiated Mukherjee-Islam (EMI)

The cd f and pd f of EMI are given, respectively, by h x  p iα F (x; α, θ, p) = , 0 ≤ x ≤ θ, θ and

(1.1.185)

αpxαp−1 , 0 < x < θ, θαp

(1.1.186)

f (x; α, θ, p) = where α, θ, p are all positive parameters.

Remark 1.63. The cd f (1.1.185) can be written as  x  pα F (x; α, θ, p) = , 0 ≤ x ≤ θ, θ which is simply a power function distribution. The power function distribution has been characterized in Hamedani’s previous work.

1.1.93

Generalized Transmuted Power Function (GTPF)

The cd f and pd f of GTPF are given, respectively, by    a ν+θ−x α F (x; α, θ, ν, λ, a,b) = 1 − × θ (   α b ) ν+θ−x 1+λ−λ 1− , ν ≤ x ≤ ν + θ, θ and

(1.1.187)

G. G. Hamedani

56

α−1    a−1 ν+θ−x α 1− × θ    b ) ν+θ−x α a(1 + λ) − λ (a + b) 1 − , θ

α f (x; α, θ, ν, λ, a, b) = θ (



ν+θ−x θ

(1.1.188)

ν < x < ν + θ, where α, θ, a, b all positive, ν ∈ R and |λ| ≤ 1 are parameters. Remark 1.64. As mentioned by Abdul-Moniem and Diab, the (1.1.187) is an special case of the one proposed by Nofal et al. (2017), which has been characterized in Hamedani (2016) before it was published in print in (2017).

1.1.94

Poisson Exponentiated Erlang-Truncated Exponential (PEETE)

The cd f and pd f of PEETE are given, respectively, by n  α o λ 1 − exp −θ 1 − e−β(1−e )x , x ≥ 0, F (x; α, β, θ, λ) = 1 − e−θ and
λ α−1 αβθ 1 − e−λ e−β(1−e )x  −β(1−eλ )x f (x; α, β, θ, λ) = 1 − e × 1 − e−θ n  α o λ exp −θ 1 − e−β(1−e )x ,

(1.1.189)

(1.1.190)

x > 0, where α, β, θ, λ are all positive parameters.  α λ Remark 1.65. Taking G1 (x) = 1 − e−β(1−e )x , x ≥ 0 in F (x; λ) of Remark 1.61, we arrive at the cd f (1.1.189).

1.1.95

Minimum Guarantee Lindley (MGL)

The cd f and pd f of MGL are given, respectively, by (
−θx ) θx 1 + θ+1 e
F (x; θ) = exp − , θx 1 − 1 + θ+1 e−θx

(1.1.191)

and

θ2 θ+1

(
−θx ) θx (1 + x) e−θx 1 + θ+1 e
f (x; θ) = exp − ,

2 θx θx 1 − 1 + θ+1 e−θx 1 − 1 + θ+1 e−θx

(1.1.192)

x > 0, where θ > 0 is a parameters.


−θx θx Remark 1.66. Taking G (x) = 1 − 1 + θ+1 e , x ≥ 0 in cd f (1.1.5), we arrive at the cd f (1.1.191).

Introduction

1.1.96

57

Inverted Beta (IB)

The cd f and pd f of IB are given, respectively, by F (x; a, b) =

xa 2 F1 (a, a + b, 1 + a, −x) , x ≥ 0, aB (a, b)

(1.1.193)

and f (x; a, b) =

xa−1 B (a, b)(1 + x)a+b

, x > 0,

(1.1.194)

where a, b are positive parameters. The cd f and pd f of IBL are given, respectively, by F (x; θ, a, b) =


a F1 (a,a+b,1+a,−x) θ 1 + θ − e−θx (1 + θ + θx) + x (1+θ)Γ(a) 2B(a,b) (1 + θ)2

(1.1.195)

,

x ≥ 0, and f (x; θ, a, b) =

1 (1 + θ)2

θ3 (1 + x) e−θx +

xa−1 (1 + θ) B (a, b)(1 + x)a+b

!

,

(1.1.196)

x > 0, where θ, a, b are all positive parameters.

1.1.97

Cubic Transmuted Pareto (CTP)

The cd f and pd f of CTP are given, respectively, by  x α   x α  x 2α 0 0 0 F (x; α, λ) = 1 − 1−λ +λ , x x x

(1.1.197)

x ≥ x0, and

f (x; α, λ) =

  x α  x 2α  αxα0 0 0 1 − 2λ + 3λ , x > x0 , xα+1 x x

(1.1.198)

where α, λ are positive parameters.

1.1.98

Zero Truncated Poisson Topp Leone Weibull (ZTPTLW)

The cd f and pd f of ZTPTLW are given, respectively, by n h i o b a 1 − exp −λ 1 − e−2x F (x; λ, a, b) = , 1 − e−λ and

x ≥ 0,

(1.1.199)

G. G. Hamedani

58

f (x; λ, a, b) =

h i b b a−1 2λab bxb−1 e−2x 1 − e−2x

1 − e−λ n h i o b a exp −λ 1 − e−2x , x > 0,

× (1.1.200)

where λ, a, b are positive parameters.

Remark 1.67. The cd f (1.1.199) is a special case of the cd f given in Remark 1.62.

1.1.99

A New (AN)

The cd f and pd f of AN are given, respectively, by   β F (x; α, β) = 1 − 1 + αxβ e−αx ,

x ≥ 0,

(1.1.201)

and

β

f (x; α, β) = α2 βx2β−1 e−αx , x > 0,

(1.1.202)

where α, β are positive parameters. Remark 1.68. The cd f (1.1.201) is a special case of the cd f given in Remark 1.22.

1.1.100

Exponentiated Kumarsawamy-G (EKw-G)

The cd f and pd f of EKw-G are given, respectively, by n oc F (x; a, b, c) = 1 − [1 − Ga (x)]b ,

x ∈ R,

(1.1.203)

and

f (x; a, b, c) = abcg (x) Ga−1 (x) [1 − Ga (x)]b−1 × n oc−1 1 − [1 − Ga (x)]b , x ∈ R,

(1.1.204)

where a, b, c are all positive parameters and G (x) is a baseline cd f with the corresponding pd f g (x). Remark 1.69. The cd f (1.1.203) is a special case of the cd f given in Remark 1.12.

Introduction

1.1.101

59

Log-Sinh Cauchy Promotion (LSCp)

The cd f and pd f of LSCp are given, respectively, by

and

n τ τ o F (x; µ, σ, ν, τ) = 1 − exp − − arctan [ν sinh(w)] , 2 π

x ∈ R+ ,

τν cosh(w) · 2 × xσπ ν sinh2 (w) + 1 o n τ τ exp − − arctan [ν sinh(w)] , x ∈ R+ 2 π

(1.1.205)

f (x; µ, σ, ν, τ) =

where µ ∈ R, σ > 0, ν > 0 and τ > 0 are parameters and w =

(1.1.206)

log(x)−µ . σ

Remark 1.70. Without Loss of Generality, we assume that µ = 0 and σ = 1.

1.1.102

Modified Beta Modified-Weibull (MBMW)

The cd f and pd f of MBMW are given, respectively, by

F (x; β, λ, a, b, c, k) =

   −1 βxk +λx B 1 + −ce +c−1 ; a, b B (a, b)

,

x ≥ 0,

(1.1.207)

and   a−1  b k k f (x; β, λ, a, b, c, k) = ca kβxk−1 + λ 1 − e−kβx −λx e−kβx −λx n  o−a−b k 1 − (1 − c) 1 − e−kβx −λx × , (1.1.208) B (a, b) x > 0, where β, λ, a, b, c, k are all positive parameters and B (z; a, b) =

1.1.103

R z a−1 (1 − t)b−1 dt. 0t

Kumaraswamy Generalized Linear Exponential (Kw-GLE)

The cd f and pd f of Kw-GLE are given, respectively, by

and

 h α ia  b θ 2 F (x; α, θ, λ, a, b) = 1 − 1 − 1 − e−( 2 x +λx) , x ≥ 0,

(1.1.209)

G. G. Hamedani

60

α−1 θ 2 x + λx × 2 αh α ia−1 θ 2 θ 2 × e−( 2 x +λx) 1 − e−( 2 x +λx) i   h α a b−1 θ 2 , 1 − 1 − e−( 2 x +λx)

f (x; α, θ, λ, a, b) = abα (θx + λ)



(1.1.210)

x > 0, where α, θ, λ, a, b are all positive parameters.

Remark 1.71. Please see Remarks 1.4, 1.11 and 1.12 for similar or more extended distributions than (1.1.209).

1.1.104

Weighted Inverse Gamma (WIG)

The cd f and pd f of WIG are given, respectively, by
γ β − c, αx F (x; α, β, c) = , x ≥ 0, Γ (β − c)

and

f (x; α, β, c) =

(1.1.211)

αβ−c 1 e−α/x, x > 0, β−c+1 Γ (β − c) x

where α, β, c are all positive parameters, Γ (α) =

(1.1.212)

R ∞ α−1 −x R e dx and γ (a, b) = 0b xa−1 e−x dx. 0 x

Remark 1.72. Castellars and Lemonte (2015) proposed the following cd f F (x; a, τ) =

1 γ (a, − log[1 − K (x)]) , x ∈ R, Γ (a)

where K (x) is a baseline cd f . Taking K (x) = 1 − e−α/x , x ≥ 0 in F (x; a, τ), we arrive at cd f (1.1.211). The cd f F (x; a, τ) has been characterized in Hamedani and Maadooliat (2017).

1.1.105

Odd Log-Logistic Generalized Half-Normal Poisson (OLLGHNP)

The cd f and pd f of OLLGHNP are given, respectively, by F (x; α, β, θ, λ)     1  = β exp  n h e −1   2Φ

and

n h
i oα λ β 2Φ θx −1 oα n h
i x λ − 1 + 2 − 2Φ θ


x λ θ



 ioα  − 1

    

,

x ≥ 0, (1.1.213)

Introduction

61

d F (x; α, β, θ, λ), x > 0, (1.1.214) dx where α > 0, β 6= 0, θ > 0, λ > 0 are parameters and Φ (x) is the cd f of the standard normal distribution. f (x; α, β, θ, λ) =

Remark 1.73. The following distribution has been characterized in Hamedani (2019) ( ! ) 1 β (G (x))α F (x; α, β) = β exp
α − 1 , x ∈ R, e −1 (G (x))α + G (x) where G (x) is a baseline cd f . Taking G (x) = 2Φ at cd f (1.1.213).

1.1.106

h


x λ θ

i

− 1, x ≥ 0 in F (x; α, β), we arrive

Topp-Leone Weighted Weibull (TLWW)

The cd f and pd f of TLWW are given, respectively, by h

−2αxγ (1+λγ )

F (x; α, γ, λ, b) = 1 − e and

ib

x ≥ 0,

,

γ

(1.1.215)

γ

f (x; α, γ, λ, b) = 2bαγ (1 + λγ ) xγ−1 e−2αx (1+λ ) × h ib−1 γ γ 1 − e−2αx (1+λ ) , x > 0,

(1.1.216)

where α, γ, λ, b are all positive parameters.

Remark 1.74. The cd f (1.1.215) is a special case of the cd f (1.1.11).

1.1.107

Burr-Hatke-G (BH-G)

The cd f and pd f BH-g are given, respectively, by θ G (x; η)   , F (x; θ, η) = 1 − 1 − log G (x; η) 

and

x ∈ R,

 θ−1     g (x; η) G (x; η) f (x; θ, η) =   2 θ 1 − log G (x; η) + 1 , 1 − log G (x; η)

(1.1.217)

(1.1.218)

x ∈ R, where θ > 0 is a parameter and G (x; η) is a baseline cd f with the corresponding pd f g (x; η).

G. G. Hamedani

62

1.1.108

Burr-Hatke Exponentiated Weibull (BHEW)

The cd f and pd f of BHEW are given, respectively, by   iθ β α 1 − 1 − e−x h F (x; α, β, θ) = 1 −
i, β α 1 − log 1 − 1 − e−x h

and

f (x; α, β, θ) =

x ≥ 0,

d F (x; α, β, θ), x > 0, dx

(1.1.219)

(1.1.220)

where α, β, θ are all positive parameters.  α −xβ Remark 1.75. Taking G (x) = 1 − e , x ≥ 0 in the cd f (1.1.217) we arrive at the cd f (1.1.17).

1.1.109

Generalized Log-Lindley (GLL)

The cd f and pd f of GLL are given, respectively, by   xσ θ (β + σ) − σxβ , F (x; σ, β, θ) = θβ + (θ − 1) σ

0 ≤ x ≤ 1,

(1.1.221)

and

f (x; σ, β, θ) =

 σ (β + σ)  θ − xβ xσ−1 , 0 < x < 1, θβ + (θ − 1) σ

(1.1.222)

where σ > 0, β ≥ 0, θ ≥ 1 are parameters.

1.1.110

Weibull Generalized Log-Logistic (WGLL)

The cd f and pd f of WGLL are given, respectively, by  h iβ  α θ F (x; α, β, θ) = 1 − exp − (1 + x ) ,

x ≥ 0,

(1.1.223)

and

h iβ−1 f (x; α, β, θ) = βθαxα−1 (1 + xα )θ−1 (1 + xα )θ ×   h iβ θ exp − (1 + xα ) , x > 0, where α, β, θ are all positive parameters.

(1.1.224)

Introduction

63

Remark 1.76. The following distributions were proposed by Pena-Ramirez et al. (2018) and Hassan and Abd-Allah (2018), respectively   β F (x; α, β, γ, λ) = 1 − exp 1 − (1 + λxγ )α , x ≥ 0, "

β  x θ −1 F (x; α, β, θ, λ, a) = 1 − exp −α 1 + λ 

#a

, x ≥ 0,

which are similar to the cd f (1.1.223). The cd f F (x; α, β, γ, λ) is that of (1.1.55) and the cd f F (x; α, β, θ, λ, a) has been characterized in Hamedani (2019).

1.1.111

Transmuted Generalized Odd Generalized Exponential-G (TGOGE-G)

The cd f and pd f of TGOGE-G are given, respectively, by  β −G(x;φ)α α 1−G(x;φ) F (x; α, β, λ, φ) = (1 + λ) 1 − e 

−λ 1−e and

−G(x;φ)α 1−G(x;φ)α

2β

x ∈ R,

,

 β−1 −G(x;φ)α −G(x;φ)α αλg (x; φ)G (x; φ)α−1 1−G(x;φ) α α 1−G(x;φ) f (x; α, β, λ, φ) = 1−e
2 e 1 − G (x; φ)α ( 2β )  −G(x;φ)α , × 1 + λ − 2λ 1 − e 1−G(x;φ)α

(1.1.225)

(1.1.226)

x ∈ R, where α, β, λ, φ are all positive parameters and G (x; φ) is a baseline cd f with the corresponding pd f g (x; φ).  β −G(x;φ)α α 1−G(x;φ) Remark 1.77. Take K (x) = 1 − e , x ∈ R and see Remark 1.27.

1.1.112

New Three Parameter Paralogistic (NTPL)

The cd f and pd f of NTPL are given, respectively, by x−θ F (x; α, θ, c) = 1 − 1 + c and





α −α

,

x ≥ 0,

     −(α+1) α2 x−θ α x−θ α f (x; α, θ, c) = 1+ , x−θ c c x > 0, where α, θ, c are all positive parameters.

(1.1.227)

(1.1.228)

G. G. Hamedani

64

Remark 1.78. The cd f (1.1.227) is similar to the cd f (1.1.47). Please also see Remark 1.17.

1.1.113

Janardan-Power Series (JPS)

The cd f and pd f of JPS are given, respectively, by    θ θαx C λe− α x 1 + θ+α 2 , F (x; α, θ, λ) = 1 − C (λ)

x ≥ 0,

(1.1.229)

and

2

f (x; α, θ, λ) =

θ λθ (1 + αx) e− α x 2 α (θ + α )

   θ θαx C0 λe− α x 1 + θ+α 2 C (λ)

,

(1.1.230)

x > 0, where α, θ, λ are all positive parameters. Remark 1.79. The cd f (1.1.229) is similar to the cd f (1.1.9). Please also see Remark 1.3.

1.1.114

Exponentiated Generalized Extended Gompertz (EGEG)

The cd f and pd f of EGEG are given, respectively, by

and

  θ a b  β γx , F (x; β, θ, γ, a, b) = 1 − 1 − 1 − e− γ (e −1)

x ≥ 0,

(1.1.231)

f (x; β, θ, γ, a, b) − βγ (eγx −1)

= abβθe



− βγ (eγx −1)

1−e

θ−1   θ a−1 − βγ (eγx −1) 1− 1−e

  θ a b−1 − βγ (eγx −1) × 1− 1− 1−e , 

(1.1.232)

x > 0, where β, θ, a, b all positive and γ ≥ 0 are parameters. Remark 1.80. The cd f (1.1.231) is similar to the cd f (1.1.34). Please also see Remark 1.12.

1.1.115

Power-Exponential Hazard Rate (P-EHR)

The cd f and pd f of P-EHR are given, respectively, by    a k+1 b cx F (x; a, b, c, k) = 1 − exp − x + (e − 1) , k+1 c

(1.1.233)

Introduction

65

and      a k+1 b cx k cx x + (e − 1) , f (x; a, b, c, k) = ax + be exp − k+1 c

(1.1.234)

x > 0, where a, b, c all non-negative and k > −1 are parameters.

1.1.116

Inverse Power Lomax (IPL)

The cd f and pd f of IPL are given, respectively, by F (x; α, β, λ) =

x−β 1+ λ

!−α

=

λxβ 1 + λxβ

!α

x ≥ 0,

,

(1.1.235)

and αβ −(β+1) x−β f (x; α, β, λ) = x 1+ λ λ

!−α−1

,

x > 0,

(1.1.236)

where α, β, λ are all positive parameters. Remark 1.81. The cd f (1.1.235) is similar to the cd f (1.1.47). Please also see Remark 1.17.

1.1.117

Exponentiated Kumaraswamy-Weibull (EK-W)

The cd f and pd f of EK-W are given, respectively, by

and

 h   i θ β a b F (x; β, θ, λ, a, b) = 1 − 1 − 1 − e−(λx) ,

x ≥ 0,

 a−1 f (x; β, θ, λ, a, b) = βθλab (λx)β−1 1 − (λx)β × h   i β a b−1 1 − 1 − e−(λx) ×  h  a ib θ−1 −(λx)β 1− 1− 1−e ,

(1.1.237)

(1.1.238)

x > 0, where β, θ, λ, a, b are all positive parameters. Remark 1.82. The cd f (1.1.237) is the same as the cd f (1.1.231). Clearly, Eissa’s work (2017) has priority over De Andrade et al. (2019).

G. G. Hamedani

66

1.1.118

Power Function Power Series (PFPS)

The cd f and pd f of PFPS are given, respectively, by F (x; α, θ, λ) = and

  x α  1 A θ , A (θ) λ

0 ≤ x ≤ λ,

αθxα−1 0   x α  A θ , 0 < x < λ, f (x; α, θ, λ) = α λ A (θ) λ

(1.1.239)

(1.1.240)

z where α, θ, λ are all positive parameters and A (θ) = ∑∞ z=1 az θ < ∞, for az ≥ 0 and θ > 0.

Remark 1.83. Ahmad et al. (2018) proposed the following cd f      λx A θ 1 − 1 + λ+1 e−λx F (x; θ, λ) = , x ≥ 0, A (θ) which was characterized in Hamedani (2019). The same characterizations can be stated for the PFPS distribution.

1.1.119

Exponentiated Negative Binomial (ENB)

The cd f and pd f of ENB are given, respectively, by

F (x; b, k, p) =

"

(1 − p) G (x)b 1 − pG (x)b

#k

=

"

γG (x)b 1 − (1 − γ) G (x)b

#k

,

x ∈ R,

(1.1.241)

and f (x; b, k, p) =

bk (1 − p)k g (x) G (x)bk−1 , x ∈ R, n ok+1 1 − pG (x)b

(1.1.242)

where b > 0, k > 0, p ∈ (0, 1) are parameters and G (x) is a baseline cd f with the corresponding pd f g (x). Remark 1.84. Hamedani, (2017) characterized the following distribution   θK (x; φ) λK (x; φ) F (x) = 1+ , x ∈ R. 1 − (1 − θ) K (x; φ) 1 − (1 − θ) K (x; φ) Taking λ = 0 and K (x; φ) = G (x)b , we arrive at the cd f (1.1.241).

Introduction

1.1.120

67

Burr-Hatke Exponential (BHE)

The cd f and pd f of BHE are given, respectively, by F (x; λ) = 1 −

e−λx , 1 + λx

x ≥ 0,

(1.1.243)

, x > 0,

(1.1.244)

and f (x; λ) =

λ (2 + λx) e−λx (1 + λx)2

λ > 0 is a parameter. Remark 1.85. Mdlongwa et al. (2017), proposed the following distribution F (x; c, k, α, β, λ) = 1 − (1 + xc )−k e−αx

β eλx

, x ≥ 0,

which has been characterized in Hamedani (2018b). The cd f (1.1.243) can be viewed as a special case of F (x; c, k, α, β, λ).

1.1.121

X Gamma Weibull (XGW)

The cd f and pd f of XGW are given, respectively, by   2 2b b 1 + θ + θxb + θ 2x e−θx F (x; θ, b) = 1 − , x ≥ 0, 1+θ and bθ2 xb−1 e−θx f (x; θ, b) = 1+θ where θ, b are positive parameters.

b



 θx2b 1+ , x > 0, 2

(1.1.245)

(1.1.246)

Remark 1.86. Ghitany et al. (2013) introduced the following distribution   1 + β + βxα F (x; α, β) = 1 − exp [−βxα ], x ≥ 0, 1+β

which is the cd f (1.1.85). Similar characterizations can be stated for (1.1.245).

1.1.122

Exponentiated Exponential Logistic (EEL)

The cd f and pd f of EEL are given, respectively, by   −λ α x/θ F (x; α, θ, λ) = 1 − 1 + e , x ∈ R,

and

(1.1.247)

G. G. Hamedani

68

f (x; α, θ, λ) =

  −λ α−1 αλex/θ x/θ 1 − 1 + e , x ∈ R,
λ+1 θ 1 + ex/θ

(1.1.248)

where α, θ, λ are all positive parameters.

1.1.123

Reduced New Modified Weibull (RNMW)

The cd f and pd f of RNMW are given, respectively, by F (x; α, β, λ) = 1 − e−α

√

√ x−β xeλx

, x ≥ 0,

(1.1.249)

and i √ √ λx 1 h f (x; α, θ, λ) = √ α + β (1 + 2λx) eλx e−α x−β xe , x > 0, 2 x

(1.1.250)

where α, β, λ are all positive parameters.

1.1.124

Extended Weibull-G (EW-G)

The cd f and pd f of EW-G are given, respectively, by (

G (x; η) F (x; α, β, λ, η) = 1 − exp −α G (x; η) 

β

 ) −λ G(x;η) G(x;η)

e

, x ∈ R,

(1.1.251)

and    G (x; η) f (x; α, β, λ, η) = αg (x; η) λ + β G (x; η) (  β  ) β−2 G(x;η) G (x; η) G (x; η) −λ G(x;η) × e exp −α , x ∈ R, G (x; η) G (x; η)β

(1.1.252)

where α, β, λ are all positive parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η).

1.1.125

Composite Generalizers of Weibull (CGW)

The cd f and pd f of CGW-1 (Exponentiated-Kumarswamy Weibull) are given, respectively, by  n h ic od ν −(αx)β F (x; α, β, ν, c, d) = 1 − 1 − 1 − 1 − e , x ≥ 0, (1.1.253)

and

Introduction

69

β

f (x; α, β, ν, c, d) = cdβναβ xβ−1 e−(αx) ×  n h ic od ν−1 h i β c−1 −(αx)β 1− 1− 1−e 1 − e−(αx) × n h io β c d−1 1 − 1 − e−(αx) ,

(1.1.254)

x > 0, where α, β, ν, c, d are all positive parameters.

Remark 1.87. Mahmoud et al. (2016) proposed the following distribution    α β b −(λx)θ F (x) = 1 − 1 − 1 − 1 − 1 − e , x ≥ 0, 

which is the same as cd f (1.1.253). The cd f F (x) of Mahmoud et al. has been characterized in Hamedani and Najibi (2016). Aryal et al. (2019) proposed 11 more distributions called: Exponentiated-Beta Weibull; Exponentiated-Transmuted Weibull; Kumarswamy-Exponentiated Weibull; KumarswamyBeta Weibull; Kumarswamy-Transmuted Weibull; Beta-Exponentiated Weibull; BetaKumarswamy Weibull; Beta-Transmuted Weibull; Transmuted-Exponentiated Weibull; Transmuted-Kumarswamy Weibull and Transmuted-Beta Weibull. Similar characterizations (stated for F (x)) can be stated for these 11 distributions as well.

1.1.126

Odd Log-Logistic Exponentiated Gumbel (OLLEGu)

The cd f and pd f of OLLEGu are given, respectively, by  ν τ 1 − (1 − e−u ) τ , x ∈ R, F (x; µ, σ, ν, τ) =  1 − (1 − e−u )ν + (1 − e−u )ντ

(1.1.255)

and

ντ−1  ν τ−1 τνue−u (1 − e−u ) 1 − (1 − e−u ) f (x; µ, σ, ν, τ) = n o2 , x ∈ R, ν τ ντ −u −u σ 1 − (1 − e ) + (1 − e )


where σ, ν, τ all positive and µ ∈ R are parameters and u = exp − x−µ σ . ν

(1.1.256)

Remark 1.88. Taking G (x) = 1 − (1 − e−u ) , x ∈ R, the cd f (1.1.107) reduces to the cd f (1.1.255). Please see Remark 1.39 as well.

G. G. Hamedani

70

1.1.127

Harris Extended Lindley (HEL)

The cd f and pd f of HEL are given, respectively, by

and

F (x; α, θ) = 1 − 

θ1/αGL (x) 1/α , x ≥ 0, 1 − θGL (x)α

(1.1.257)

θ1/αλ2 (1 + x) e−λx (1.1.258)  1+1/α , x > 0, (1 + λ) 1 − θGL (x)α   e−λx , x ≥ 0, is survival function where α, θ are positive parameters and GL (x) = 1+λ+λx 1+λ of the Lindley distribution. f (x; α, θ) =

Remark 1.89. Pinho et al. (2016) proposed the following distribution "

F (x) = 1 −

θG(x)k 1 − θG (x)k

#1/k

, x ≥ 0,

which has been characterized in Hamedani and Najibi (2016). The characterizations of the cd f (1.1.257) are similar to those of F (x) given above.

1.1.128

Odd Burr III Weibull (OBIIIW)

The cd f and pd f of OBIIIW are given, respectively, by F (x; α, β, c, k) =

(

1+

and

α

e−(x/β) α 1 − e−(x/β)

!c )−k

, x ≥ 0,

h ic  α−1 −(x/β)α e α x × f (x; α, β, c, k) = ck  α c+1 β β 1 − e−(x/β) ( !c )−k−1 α e−(x/β) 1+ , x > 0, α 1 − e−(x/β)

(1.1.259)

(1.1.260)

where α, β, c, k are all positive parameters. Remark 1.90. Jamal et al. (2017) introduced the following distribution    −k 1 − K (x; θ) c F (x; c, θ, k) = 1 + , x ∈ R, K (x; θ) α

where K (x; θ) is a baseline cd f . Taking K (x; γ) = 1 − e−(x/β) , x ≥ 0, in F (x; c, θ, k), we arrive at the cd f (1.1.259). The cd f F (x; c, θ, k) has been characterized in Hamedani (2018b).

Introduction

1.1.129

71

Power Lindley Generated (PLG)

The cd f and pd f of PLG are given, respectively, by  α   G(x;η) α  θ G(x;η)  −θ G(x;η)  G(x;η) , x ∈ R, F (x; α, θ, η) = 1 − 1 + e θ+1

(1.1.261)

and

f (x; α, θ, η) =

αθ2 g (x; η) G (x; η)α−1

× (θ + 1) G (x; η)α+1      α G (x; η) α −θ G(x;η) G(x;η) 1+ e , x ∈ R, G (x; η)

(1.1.262)

where α, θ are positive parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η). Remark 1.91. Ahsan ul Haq et al. (2018) proposed the following distribution     G (x; η) G (x; η) exp − , x ∈ R, F (x; β, η) = 1 − 1 + βG (x; η) βG (x; η) where G (x; θ) is a baseline cd f . For α = 1, the cd f (1.1.261) reduces to the cd f F (x; β, η), which has been characterized in Hamedani (2019). Similar characterizations can be stated for the cd f (1.1.261).

1.1.130

Inverse Gompertz (IG)

The cd f and pd f of IG are given, respectively, by   β − αβ e x −1

F (x; α, β) = e

, x ≥ 0,

(1.1.263)

and α − αβ f (x; α, β) = 2 e x where α, β are positive parameters.

  β e x −1 + βx

, x > 0,

(1.1.264)

Remark 1.92. Oguntunde and Adejumo (2015) proposed the following distribution  α λ F (x; α, γ, λ) = 1 − 1 − e−γ( x ) , x ≥ 0.

For α = 1, the cd f F (x; α, γ, λ) reduces to the cd f (1.1.263). F (x; α, γ, λ) has been characterized in Hamedani (2018b).

The distribution

G. G. Hamedani

72

1.1.131

Hyperbolic Sine Rayleigh (HS-R)

The cd f and pd f of HS-R are given, respectively, by F (x; α, σ) = and f (x; α, σ) =

2eα (eα − 1) 4ασx

     −σx2 cosh α 1 − e − 1 , x ≥ 0, 2

(1.1.265)

   α−σx2 −σx2 e sinh α 1 − e , x > 0, 2

(1.1.266)

(eα − 1) where α, σ are positive parameters.

1.1.132

Type II Topp-Leone Generated (TIITL-G)

The cd f and pd f of TIITL-G are given, respectively, by  α F (x; α) = 1 − 1 − G2 (x) , x ∈ R,

(1.1.267)

 α−1 f (x; α) = 2αg (x) G (x) 1 − G2 (x) , x ∈ R,

(1.1.268)

and

where α > 0 is a parameter and G (x) is a baseline cd f with the corresponding pd f g (x). Remark 1.93. Mahmoud et al. (2015), proposed the following distribution n h
β ia ob F (x; α, β, a, b, η) = 1 − 1 − 1 − 1 − (K (x; η))α , x ∈ R,

which is a more general form of the cd f (1.1.267). Please see Remark 1.12 for more detail in this regard.

1.1.133

Exponentiated Inverse Rayleigh (EIR)

The cd f and pd f of EIR are given, respectively, by  α −(σ/x)2 F (x; α, σ) = 1 − 1 − e , x ≥ 0,

(1.1.269)

and

α−1 2ασ2 −(σ/x)2  −(σ/x)2 e 1 − e , x > 0, x3 where α, σ are positive parameters. f (x; α, σ) =

(1.1.270)

Remark 1.94. Mansour et al. (2018) proposed the following distribution     a b λ α −( σx ) F (x; α, σ, λ, a, b) = 1 − 1 − 1 − 1 − e , x ≥ 0, which is clearly way more general than that of the cd f (1.1.269). Mansour et al. distribution has been characterized in Hamedani (2019).

Introduction

1.1.134

73

Exponentiated New Weighted Weibull (ENWW)

The cd f and pd f of ENWW are given, respectively, by

and

h iβ θ θ F (x; α, β, θ, λ) = 1 − e−(αx +α(λx) ) , x ≥ 0,   θ θ f (x; α, β, θ, λ) = β 1 + λθ αθxθ−1 e−(αx +α(λx) ) × h iβ−1 θ θ 1 − e−(αx +α(λx) ) , x > 0,

(1.1.271)

(1.1.272)

where α, β, θ, λ are positive parameters.

Remark 1.95. Nofal et al. (2018) proposed the following distribution h iδ  h iδ  −(αxθ +γxβ ) −(αxθ +γxβ ) F (x; α, β, γ, δ, λ) = 1 − e 1+λ−λ 1−e ,

which clearly is way more general than the cd f (1.1.271). The cd f F (x; α, β, γ, δ, λ) has been characterized in Hamedani (2017).

1.1.135

Beta Transmuted Weighted Exponential (BTWE)

The cd f and pd f of BTWE are given, respectively, by 1 F (x; α, β, λ, a, b) = B (a, b)

Z [1−e−α(β+1)x ][1+e−α(β+1)x ] 0

t a−1 (1 − t)b−1 dt , x ≥ 0, (1.1.273)

and d F (x; α, β, λ, a, b), x > 0, dx where α, β, a, b all positive and λ (|λ| ≤ 1) are parameters. f (x; α, β, λ, a, b) =

(1.1.274)

Remark 1.96. In Remark 1.15, we mentioned that the Generalized Class of distributions with the following cd f was introduced in (2015) Q(x) 1 t α−1 (1 − t)β−1 dz, x ∈ R, B (a, b) 0    where Q (x) is a cd f . Taking Q (x) = 1 − e−α(β+1)x 1 + e−α(β+1)x , x ≥ 0, in F (x; α, β), we arrive at (1.1.273). The distribution F (x; α, β) has been characterized in Hamedani (2016).

F (x; α, β) =

Z

G. G. Hamedani

74

1.1.136

Odd Lindley Exponentiated Weibull (OLi-EW)

The cd f and pd f of Oli-EW are given, respectively, by

n

h

F (x; α, β, a) = 1 − 1 − 1 − e−x

 n h iα o −xβ i o a + 1 − 1 − e −1 α β 1+a

h i  h i  β α β α −a 1−e−x / 1− 1−e−x

e

× (1.1.275)

,

x ≥ 0, and f (x; α, β, a) =

d F (x; α, β, a), x > 0, dx

(1.1.276)

where α, β, a are all positive parameters. 1 Remark 1.97. The cd f reported in Refaie is missing the coefficient 1+a , which is corrected in (1.1.275). The cd f (1.1.275) is exactly the cd f (1.1.87), so Refaie’s proposed distribution is not new.

1.1.137

Modified Weibull-G (MW-G)

The cd f and pd f of MW-G are given, respectively, by h    γ i G(x;η) − β G(x;η) +λ G(x;η) G(x;η)

F (x; β, λ, γ, η) = 1 − e

, x ∈ R,

(1.1.277)

and "

G (x; η) f (x; β, λ, γ, η) = β + λγ G (x; η) 

g (x; η) −  2 e G (x; η)

γ−1 #

×

h    γ i G(x;η) β G(x;η) +λ G(x;η) G(x;η)

, x ∈ R,

(1.1.278)

where β, λ, γ, η are all positive parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η). Remark 1.98. a more general case of (1.1.277) was proposed by Hassan and Hemeda (2016)as follows   d  b  G(x;η) +a G(x;η) − c G(x;η) G(x;η)

F (x; a, b, c, d) = 1 − e

, x ∈ R.

The cd f F (x; a, b, c, d) has been characterized in Hamedani and Safavimanesh (2017).

Introduction

1.1.138

75

Compound Gamma and Lindley (GaL)

The cd f and pd f of GaL are given, respectively, by F (x; α, β) =

1 xα [x (1 + β) + (1 + α + β) β] , 1+β (β + x)α+1

x ≥ 0,

(1.1.279)

αβ2 xα−1 (1 + α + β + x) , 1+β (β + x)α+2

x > 0,

(1.1.280)

and f (x; α, β) =

where α, β are positive parameters.

1.1.139

Odd Weibull (OW)

The cd f and pd f of OW are given, respectively, by

and

n h  α ioν 1 − exp − βx F (x; α, β, ν) = n h  α ioν n h  α ioν , 1 − exp − βx + exp − βx

x ≥ 0,

(1.1.281)

f (x; α, β, ν) h  α i hn h  α io n h  α ioiν−1 ναxα−1 exp − βx 1 − exp − βx exp − βx = , n h  α ioν n h  α ioν 2 x x α β 1 − exp − β + exp − β

x > 0,

(1.1.282)

where α, β, ν are all positive parameters. Remark 1.99. Alizadeh et al. (2017) considered the following distribution i   β α λ β 1 − 1 + 1+λ x e−λx  iα h  iα , x ≥ 0. F (x; α, β, λ) = h  β β λ β λ β 1 − 1 + 1+λ x e−λx + 1 + 1+λ x e−λx h

As far as the characterizations of cd f (1.1.281) are concerned, they will be similar to those of F (x; α, β, λ), which has been characterized in Hamedani (2017).

G. G. Hamedani

76

1.1.140

Topp-Leone Generalized Inverted Kumarswamy (TLGIKw)

The cd f and pd f of TLGIKw are given, respectively, by ) θ   −αβ 2  , F (x; α, β, θ, λ) = 1 − 1 − 1 − 1 + xλ 

and

(

λ−1

f (x; α, β, θ, λ) = 2αβθλx (





1 − 1 − 1 + xλ

−α β

x ≥ 0,

(1.1.283)

 −α−1   −α β−1 λ λ 1+x 1− 1+x ×

) ( ) θ−1   −α β 2  , 1 − 1 − 1 − 1 + xλ

(1.1.284)

x > 0, where α, β, θ, λ are all positive parameters.

Remark 1.100. Paranaíba et al. (2013) proposed a distribution with the cd f given below     x γ −k α b , x ≥ 0, F (x; α, γ, k, a, b) = 1 − 1 − 1 − 1 + α

where all the parameters are positive. The cd f (1.1.283) is equal to (F (x; α, γ, k, a, b))θ for α = 2. The cd f F (x; α, γ, k, a, b) has been characterized in Hamedani (2016).

1.1.141

Marshall-Olkin Burr X (MOBX)

The cd f and pd f of MOBX are given, respectively, by

and

n o 2 θ 1 − e−[λHR (x)]  F (x; α, θ, λ, η) = n o , 2 θ −[λH (x)] R 1−α 1− 1−e

x ≥ 0,

n o 2 2 θ−1 2λ2 θhR (x) HR (x) e−[λHR (x)] 1 − e−[λHR (x)] , f (x; α, θ, λ, η) =   n o 2 2 θ 1 − α 1 − 1 − e−[λHR (x)]

(1.1.285)

(1.1.286)

x > 0, where α, θ, λ are all positive parameters and HR (x) = − log [1 − R (x; η)], R (x; η) is d a baseline cd f and hR (x) = dx HR (x).

Introduction

77

Remark 1.101. Hamedani and Risti´c (Marshall-Olkin Extended Distribution, 2016) proposed their distribution based on the cd f given below δ αK (x) , x ∈ R, F (x; α, δ) = 1 − 1 − αK (x) where K (x) is a baseline cd f . Taking δ = 1 in F (x; α, δ), we arrive at 

F (x; α, 1) =

K (x) , x ∈ R. 1 − α(1 − K (x))

n o 2 θ Letting K (x) = 1 − e−[λHR (x)] , x ≥ 0 in F (x; α, 1), we have the cd f (1.1.285). The cd f F (x; α, δ) has been characterized in Hamedani and Risti´c (2016).

1.1.142

New Extended Generalized Burr III (NEGBIII)

The cd f and pd f of NEGBIII are given, respectively, by  
−k  F (x; c, k) = FT QY 1 + x−c ,

x ≥ 0,

(1.1.287)

and

f (x; c, k) = ckx−c−1 1 + x−c


−k−1

   −k ft QY (1 + x−c )    , x > 0, fy QY (1 + x−c )−k

(1.1.288)

where c > 0, k > 0 are all positive parameters. For the definitions of QY , FT , ft and f y please see the authors paper (Jamal et al. (2018)).

1.1.143

Generalized Inverse Weibull-Generalized Inverse Weibull (GIW-GIW)

The cd f and pd f of GIW-GIW are given, respectively, by "   # α  x b β F (x; α, β, θ, a, b, c) = 1 − exp −θ , c a

x ≥ 0,

(1.1.289)

and

  αβθb  x b−1 α  x b β−1 f (x; α, β, θ, a, b, c) = × ac a c a "  # β α  x b exp −θ , x > 0, c a

(1.1.290)

where α, β, θ, a, b, c are all positive parameters.

Remark 1.102. It is clear that the cd f (1.1.289) is that of an extended exponential distribution. It is indeed a special case of the cd f given in Remark 1.7.

G. G. Hamedani

78

1.1.144

Exponentiated Burr XII Power Series (EBXIIPS)

The cd f and pd f of EBXIIPS are given, respectively, by  h iα  C λ − λ 1 − (1 + xc )−k , F (x; α, λ, c, k) = 1 − C (λ) and f (x; α, λ, c, k) =

×

αλckxc−1 h

1 − (1 + xc )−k

(1.1.291)

iα−1

(1 + xc )k+1  h iα  C0 λ − λ 1 − (1 + xc )−k C (λ)

x ≥ 0,

, x > 0,

(1.1.292)

n where α, c, k positive, λ ∈ (0, s) are parameters and C (λ) = ∑∞ n=1 an λ , an ≥ 0 and λ ∈ (0, s) is chosen such that C (λ) is finite.

Remark 1.103. As far as the characterizations are concerned, the cd f (1.1.291) is a special case of the cd f given in Remark 1.35 with the obvious choice of G (x) .

1.1.145

Burr XII Weibull Logarithmic (BWL)

The cd f and pd f of BWL are given, respectively, by   β log 1 − θ (1 + xc )−k e−αx F (x; α, β, θ, c, k) = 1 − , log(1 − θ) and

x ≥ 0,

d F (x; α, β, θ, c, k), x > 0, dx where α, β, c, k positive, θ ∈ (0, 1) are parameters. f (x; α, β, θ, c, k) =

(1.1.293)

(1.1.294)

Remark 1.104. As far as the characterizations are concerned, the cd f (1.1.293) is a special case of the cd f given in Remark 1.34 with the obvious choice of G1 (x).

1.1.146

Odd Log-Logistic Geometric Normal (OLLGN)

The cd f and pd f of OLLGN are given, respectively, by
Φτ x−µ σ F (x; µ, σ, τ, ν) =

τ , Φτ x−µ + (1 − ν) 1 − Φ x−µ σ σ

x ∈ R,

(1.1.295)

and


τ−1 x−µ

τ−1 (1 − ν) τφ x−µ 1 − Φ x−µ σ Φ σ σ f (x; µ, σ, τ, ν) = , x ∈ R, n o2  x−µ
x−µ
τ τ Φ σ + (1 − ν) 1 − Φ σ

(1.1.296)

Introduction

79

where µ ∈ R, σ > 0, τ > 0 and ν ∈ (0, 1) are parameters. Remark 1.105. Without Loss of Generality,we assume µ = 0 and σ = 1.

1.1.147

Marshall-Olkin Extended Flexible Weibull (MOEFW)

The cd f and pd f of MOEFW are given, respectively, by   a exp −bκαβ (x)  , F (x; α, β, a, b) = 1 − 1 − (1 − a) exp −bκαβ (x)

x ≥ 0,

(1.1.297)

and


  ab α + β/x2 exp αx + β/x − καβ (x) f (x; α, β, a, b) = ×    2 1 − (1 − a) exp −bκαβ (x)   exp − (b − 1) καβ (x) , x > 0,

(1.1.298)

where α, β, a, b are all positive parameters and καβ (x) = exp (αx − β/x).

1.1.148

Topp-Leone Inverse Weibull (TLIW)

The cd f and pd f of TLIW are given, respectively, by  n o2 α −β/xγ F (x; α, β, γ) = 1 − 1 − e ,

x ≥ 0,

(1.1.299)

and

f (x; α, β, γ) =

o n o2 α−1 2αβγ −β/xγ n −β/xγ −β/xγ e 1 − e 1 − 1 − e , xγ+1

(1.1.300)

x > 0, where α, β, γ are all positive parameters. Remark 1.106. Mansour et al. (2018) proposed the following distribution    α a b λ −( σx ) F (x; α, σ, λ, a, b) = 1 − 1 − 1 − 1 − e , x ≥ 0. Taking b = 1, in F (x; α, σ, λ, a, b), we arrive at a more general distribution than (1.1.299). Mansour et al. distribution has been characterized in Hamedani (2019).

80

1.1.149

G. G. Hamedani

Odd Log-Logistic Marshall-Olkin Power Lindley (OLLMOPL)

The cd f and pd f of OLLMOPL are given, respectively, by  i h  β α λ β x e−λx (1 + λ)−1 1 − 1 + 1+λ  iα  α , x ≥ 0, (1.1.301)  F (x; α, β, θ, λ) = h λ β λ β x e−λxβ + θ 1 + 1+λ x e−λαxβ 1 − 1 + 1+λ

and

d F (x; α, β, θ, λ), x > 0, dx where α, β, θ, λ are all positive parameters. f (x; α, β, θ, λ) =

(1.1.302)

Remark 1.107. Alizadeh et al. (2017) considered the following distribution h   i β α λ β 1 − 1 + 1+λ x e−λx F (x; α, β, λ) = h , x ≥ 0.   iα   β β λ β λ β 1 − 1 + 1+λ x e−λx + 1 + 1+λ x e−λx

As far as the characterizations of the cd f (1.1.301) are concerned, they will be similar to those of F (x; α, β, λ), which has been characterized in Hamedani (2017).

1.1.150

Topp-Leone Lomax (TLLo)

The cd f and pd f of TLLo are given, respectively, by h ia F (x; a, b, c) = 1 − (1 + cx)−2b , x ≥ 0,

(1.1.303)

and

h ia−1 f (x; a, b, c) = 2abc (1 + cx)−(2b+1) 1 − (1 + cx)−2b , x > 0,

(1.1.304)

where a, b, c are all positive parameters.

Remark 1.108. Gómez et al. (2014) proposed the following distribution h iα F (x; α, θ, λ) = 1 − (1 + λx)−θ , x ≥ 0,

which is the same as the cd f (1.1.303). Furthermore, a generalization of Gómez et al. was introduced by Mead (2015) which has been characterized in Hamedani and Maadooliat (2017).

Introduction

1.1.151

81

Exponentiated Topp-Leone (ETL)

The cd f and pd f of ETL are given, respectively, by

and

h  x ν  x ν iβ F (x; β, ν, b) = 1 − 1 − 2− , 0 ≤ x ≤ b, b b 2νβ  x x  x ν−1 f (x; β, ν, b) = 1− 2− × b b b b h  x ν  x ν iβ−1 2− , 0 < x < b, 1− b b

(1.1.305)

(1.1.306)

where β, ν, b are all positive parameters.

Remark 1.109. Cordeiro et al. (2013) proposed the following distribution  β F (x; α, β) = 1 − {1 − G (x)}α , x ∈ R,

where α, β
are positive
ν parameters and G (x) is a cd f with support in R. Taking, ν G (x) = bx 2 − bx , 0 ≤ x ≤ b, we arrive at the cd f (1.1.305). The cd f F (x; α, β) has been characterized in Hamedani (2016).

1.1.152

Topp-Leone Generator (TLG)

The cd f and pd f of TLG are given, respectively, by F (x; α) = G (x)α (2 − G (x))α , x ∈ R ,

(1.1.307)

and f (x; α) = 2αg (x) (1 − G (x))G (x)α−1 (2 − G (x))α−1 , x ∈ R,

(1.1.308)

where α > 0 is a parameter and G (x) is a baseline cd f with the corresponding pd f g (x). Remark 1.110. Nassar et al. (2015) proposed the following distribution h i a b F (x; λ, a, b, ϕ) = K (x; ϕ) (1 + λ) − λK (x; ϕ) , x ∈ R,

where λ (|λ| ≤ 1) , a > 0, b > 0 are parameters and K (x; ϕ) is a baseline cd f . The characterizations of the cd f (1.1.307) will be similar to those of F (x; λ, a, b, ϕ), which have appeared in Hamedani (2016).

G. G. Hamedani

82

1.1.153

Topp-Leone Generated q-Exponential (TLG-qE)

The cd f and pd f of TLG-qE are given, respectively, by oα n 2−q , x ≥ 0, F (x; α, λ, q) = 1 − [1 − (1 − q) λx]2( 1−q )

(1.1.309)

and

3−q

f (x; α, λ, q) = 2αλ (2 − q) [1 − (1 − q) λx] 1−q n oα−1 2−q × 1 − [1 − (1 − q) λx]2( 1−q ) , x > 0,

(1.1.310)

where α > 0, λ > 0 and q ∈ (1, 2) are parameters o n 2−q Remark 1.111. Taking G (x) = 1 − [1 − (1 − q) λx]2( 1−q ) , x ≥ 0, the cd f (1.1.309) will

become F (x; α, λ, q) = (G (x))α which has been characterized in Hamedani (2019). The authors have the conditions q 6= 0 , q < 2. For q ≤ 1, (1.1.309) will not be a cd f . The correct condition is q ∈ (1, 2).

1.1.154

Odd Hyperbolic Cosine KG (OHC-KG)

The cd f and pd f of OHC-KG are given, respectively, by    G (x) 2ea F (x; a) = 2a sinh aK , x ∈ R, e −1 1 − G (x)

(1.1.311)

and

  2aea g (x) G (x) f (x; a) = 2a k e − 1 (1 − G (x))2 1 − G (x)    G (x) × cosh aK , x ∈ R, 1 − G (x)

(1.1.312)

where a > 0 is a parameter and G and K are cd f 0 s with the corresponding pd f 0 s g and k.

1.1.155

Generalized Gudermannian (GG)

The cd f and pd f of GG are given, respectively, by F (x; β) = and

  2 arctan eβx , x ∈ R, π

f (x; β) = where β > 0 is a parameter.

2βeβx
, x ∈ R, π 1 + e2βx

(1.1.313)

(1.1.314)

Introduction

1.1.156

83

New Extended Alpha Power Transformed (NEAPT)

The cd f and pd f of NEAPT are given, respectively, by F (x; α, η) = and

αG(x;η) − eαG(x;η) , x ∈ R, α − eα

  g (x; η) (logα)αG(x;η) − αeαG(x;η) , x ∈ R, f (x; α, η) = α − eα

(1.1.315)

(1.1.316)

where α > 0 (α 6= e) , η > 0 are parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η).

1.1.157

Exponentiated Odd Log-Logistic-G (EOLL-G)

The cd f and pd f of EOLL-G are given, respectively, by F (x; a, b, η) = 

and

G (x; η)ab G (x; η)a + G (x; η)a

b , x ∈ R,

(1.1.317)

d F (x; a, b, η), x ∈ R, (1.1.318) dx where a > 0, η > 0 are parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η) which may depend on the parameter vector η. f (x; a, b, η) =

Remark 1.112. The following cd f has been introduced a while back G (x; η)α F (x; α, θ, τ) = , x ∈ R, G (x; η)α + G (x; η)α where α, θ are positive parameters and G (x; τ) is the baseline cd f which may depend on the parameter vector τ. For b = 1, the cd f (1.1.317) will reduce to the cd f F (x; α, θ, τ), which has been characterized in Hamedani (2016). Clearly, similar characterizations can be stated for the following cd f (1.1.317).

1.1.158

Zero Truncated Poisson Topp Leone Weibull (ZTPTLW)

The cd f and pd f of ZTPTLW are given, respectively, by

F (x; a, b, λ) = and

n h  ia o 1  b 1 − exp −λ 1 − exp −2x , x ≥ 0, 1 − e−λ

(1.1.319)

G. G. Hamedani

84

b  ia−1 2λabxb−1e−2x h b 1 − exp −2x 1 − e−λ  n h  ia o × 1 − exp −λ 1 − exp −2xb ,

f (x; a, b, λ) =

(1.1.320)

x > 0, where a, b, λ are all positive parameters.

Remark 1.113. Almamy (2019) proposed the following distribution (see cd f (1.1.111)) F (x; λ, υ, α) =

  2α υ −λ 1−e−x

1−e 1 − e−λ

, x ≥ 0,

x > 0, where λ, υ, α are all positive parameters. The F (x; λ, υ, α) is exactly that of (1.1.319). Furthermore, Nadarajah et al. (2009) proposed the following distribution 1 − e−λG(x) , x ∈ R, 1 − e−λ   b a where G (x) is a baseline cd f . Taking G (x) = 1 − e−2x , x ≥ 0, in F (x), we arrive at cd f (1.1.319). The cd f F (x) has been characterized in Hamedani (2016). F (x) =

1.1.159

Type II Generalized Topp-Leone-G (TIIGTL-G)

The cd f and pd f of TIIGTL-G are given, respectively, by  α F (x; α, β, ζ) = 1 − 1 − G (x; ζ)2β , x ∈ R,

(1.1.321)

and

 α−1 f (x; α, β, ζ) = 2αβg (x; ζ) G (x; ζ)2β−1 1 − G (x; ζ)2β , x ∈ R,

(1.1.322)

where α > 0, β > 0 are parameters and G (x; ζ) is a baseline cd f with the corresponding pd f g (x; ζ), which may depend of the parameter vector ζ. Remark 1.114. Rezaei et al. (2016) proposed the following distribution n o b θ F (x; a, b, θ, ξ) = 1 − 1 − {1 − (1 − G (x; ξ))a } , x ∈ R,

which clearly is more general than cd f (1.1.321). The cd f F (x; a, b, θ, ξ) has been characterized in Hamedani (2017).

Introduction

1.1.160

85

Burr XII Exponentiated Weibull (BXIIEW)

The cd f and pd f of BXIIEW are given, respectively, by  α −β   b a  −x  1−e  
, x ≥ 0, F (x; α, β, a, b) = 1 − 1 + a b   1 − 1 − e−x     

and

 aα−1 −xb 1 − e b f (x; α, β, a, b) = αβabxb−1e−x 
 b a α+1 1 − 1 − e−x  a α −(1+β)     −xb   1−e  1+
, b a   1 − 1 − e−x  

(1.1.323)

(1.1.324)

x > 0, where α, β, a, b are all positive parameters.

Remark 1.115. Alizadeh et al. (2019) proposed the following distribution  − 1 G (x; η) α β F (x; α, β, η) = 1 − 1 + β , x ∈ R. G (x; η)   b a Taking G (x; η) = 1 − e−x , x ≥ 0, we arrive, more or less, at cd f (1.1.323). The cd f F (x; α, β, η) has been characterized in Hamedani (2019). 

1.1.161



Generalized Odd Log-Logistics Inverse Weibull (GOLLIW)

The cd f and pd f of GOLLIW are given, respectively, by −1 b

and

e−αθc(ax ) F (x; α, θ, c, a, b) = h iα , x ≥ 0, −1 )b −1 )b −αθc(ax −αθc(ax e + 1−e

−1 αθcbab x−(b+1)e−αθc(ax )

f (x; α, θ, c, a, b) =

n

e−αθc(ax

−1 )b

b

  b α−1 −αθc(ax−1 ) 1−e

h iα o2 −1 b + 1 − e−αθc(ax )

where α, θ, c, a, b are all positive parameters.

, x > 0,

(1.1.325)

(1.1.326)

G. G. Hamedani

86

Remark 1.116. Cordeiro et al. (2016) introduced the following distribution F (x; α, θ, τ) = b

(G (x; τ))αθ h iα , x ∈ R. (G (x; τ))αθ + 1 − (G (x; τ))θ

−1 Taking G (x; τ) = e−c(ax ) , x ≥ 0, the cd f F (x; α, θ, τ) reduces to the cd f (1.1.325). The cd f F (x; α, θ, τ) has been characterized in Hamedani (2019), so similar characterizations can be stated for the cd f (1.1.325).

1.1.162

Centered Skew-Normal Birnbaum-Saunders (CSNBS)

The cd f and pd f of CSNBS are given, respectively, by F (x; α, β, µ, σ, λ) =

Z x 0

f (u; α, β, µ, σ)du , x ≥ 0,

(1.1.327)

and f (x; α, β, µ, σ, λ) = 2φ [ax;µ,σ (α, β)] Φ [λax;µ,σ (α, β)] Ax;σ (α, β), x > 0,

(1.1.328)

where α, β, σ, λ all positive , µ ∈ R are parameters  and ax;µ,σ (α, β) = µ + σφax (α, β), q q β x d Ax;σ (α, β) = σAx (α, β), ax (α, β) = α1 and Ax (α, β) = dx ax (α, β) = β− x 1 x−3/2 (x + β). 2αβ1/2

1.1.163

Logarithmic Kumarswamy (LKu)

The cd f and pd f of LKu are given, respectively, by

F (x; α, β, γ, η) = 1 −

h n  γ o i log eα − 1 − 1 − G (x; η)β (eα − 1) α

, x∈R

(1.1.329)

Expansion , and  γ−1 γβ (eα − 1) g (x; η) G (x; η)β−1 1 − G (x; η)β f (x; α, β, γ, η) = h n  γo i , x ∈ R, β α α α e − 1 − 1 − G (x; η) (e − 1)

(1.1.330)

where α, β, γ, η are all positive parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η).

Introduction

1.1.164

87

Marshall-Olkin Odd Lindley G (MOOL-G)

The cd f and pd f of MOOL-G are given, respectively, by F (x; p, α, φ) = and f (x; p, α, φ) =

KOL (x; α, φ) , x ∈ R, 1 − K OL (x; α, φ)

(1.1.331)

d F (x; p, α, φ), x ∈ R, dx

(1.1.332) h i G1 (x;φ) α+G1 (x;φ) where p ∈ (0, 1), α > 0 are parameters, KOL (x; α, φ) = 1 − (1+α)G exp −α , G1 (x;φ) 1 (x;φ) x ∈ R and G1 (x; φ) is a baseline cd f which may depend on the parameter vector φ. Remark 1.117. The following distribution has been characterized in Hamedani and Najibi (2016) F (x) = 1 −

"

θG (x)k 1 − θG (x)k

#1/k

,

x ∈ R.

Taking k = 1 and p = 1 − θ, G (x) = KOL (x; α, φ) in F (x), we arrive at the cd f (1.1.331).

1.1.165

Generalized Transmuted Fréchet (GTF)

The cd f and pd f of GTF are given, respectively, by h  i −α −α F (x; α, β, λ, a, b) = e−aβx 1 + λ 1 − e−bβx , x ≥ 0,

(1.1.333)

  # −α a (1 + λ) 1 − e−aβx

(1.1.334)

and

−(α+1)

f (x; α, β, λ, a, b) = αβx

"

− (a + b) e−(a+b)βx

−α

,

x > 0, where α, β, a, b all positive and λ ∈ [−1, 1] are parameters. Remark 1.118. Nofal and Ahsanullah (2018) proposed the following distribution.

        α β α β F (x; α, β, λ, a, b) = exp −a 1 + λ − λ exp −b , x ≥ 0. x x where α, β, a, b all positive and λ (|λ| ≤ 1) are parameters, which is clearly the same as the cd f (1.1.333).

G. G. Hamedani

88

Khan et al. (2017) proposed the following distribution   " (  β )#φ   α 1 F (x; α, β, γ, φ, λ) = 1 − 1 − exp − − γ ×   x x   " (  β )#φ   1 α , x ≥ 0. 1 + λ 1 − exp − − γ   x x

Clearly F (x; α, β, γ, φ, λ) is more general than cd f (1.1.333). The distribution F (x; α, β, γ, φ, λ) has been characterized in Hamedani (2018a). We would like to mention that the cd f (1.1.333) (and in turn F (x; α, β, γ, φ, λ)) is a first step generalization of the Transmuted Fréchet distribution of Mahmoud and Mandouh (2013).

1.1.166

Weibull Generalized Exponentiated Weibull (WGEW)

The cd f and pd f of WGEW are given, respectively, by n h iα oθ γ  −xβ 1 − 1 − 1 − e    F (x; α, β, γ, θ) = 1 − exp −  n  α oθ   , x ≥ 0, β 1 − 1 − e−x  

and

iγα−2α−1 −xβ 1 − e β f (x; α, β, γ, θ) = αβγθxβ−1 e−x n   oθγ+1 × β α −x 1− 1−e   n h iα oθ γ −xβ  1− 1− 1−e  exp −  n  α oθ   , β 1 − 1 − e−x

(1.1.335)

h

(1.1.336)

x > 0, where α, β, γ, θ are all positive parameters.

Remark 1.119. Hassan and Elgarhy (2019) proposed the following distribution "   #a −α

F (x; α, β, a, η) = 1 − e

K(x;η) 1−K(x;η)

β

, x ∈ R.

n h i oθ β α Taking K (x; η) = 1− 1 − 1 − e−x , F (x; α, β, a, η) reduces to the cd f (1.1.335). The cd f F (x; α, β, a, η) has been characterized in Hamedani (2019).

Introduction

1.1.167

89

Odd Log-Logistic Poisson-G (OLLP-G)

The cd f and pd f of OLLP-G are given, respectively, by h βG(x;ψ) iα e

F (x; α, β, ψ) = h

and

eβG(x;ψ) −1 eβ −1

iα

−1 eβ −1

h iα , x ∈ R, βG(x;ψ) + 1 − e eβ −1−1

 α−1  β α−1 αβg(x; ψ)eβG(x;ψ) eβG(x;ψ) − 1 e − eβG(x;ψ) , f (x; α, β, ψ) = iα o2
nh eβG(x;ψ) −1 iα h eβG(x;ψ) −1 β e −1 + 1 − eβ−1 eβ −1

(1.1.337)

(1.1.338)

x ∈ R, where α, β are positive parameters and G(x; ψ) is a baseline cd f with the corresponding pd f g(x; ψ), which may depend on the parameter vector ψ. Remark 1.120. Alizadeh et al. (2017) proposed the following distribution F (x) =

G (x; γ)α , x ∈ R, G (x; γ)α + βG (x; γ)α βG(x;ψ)

where G (x; γ) is a baseline cd f . Taking G (x; ψ) = e eβ −1−1 , x ∈ R, in the above F (x) , we arrive at the cd f (1.1.337). The characterizations of the cd f (1.1.337) are similar to those of F (x) which have appeared in Hamedani (2019).

1.1.168

New Lindley Exponential (NLE)

The cd f and pd f of NLE are given, respectively, by   θ  x  −(x/λ)θ e , x ≥ 0, F (x; θ, λ) = 1 − 1 + θ+1 λ

(1.1.339)

and

f (x; θ, λ) =

 x i θ θθ2 h 1+ e−(x/λ) , x > 0, λ (θ + 1) λ

(1.1.340)

where α, λ are positive parameters.

Remark 1.121. Ghitany et al. (2013) introduced the following distribution   β α F (x; α, β) = 1 − 1 + x exp [−βxα ] , x ≥ 0, β+1

which is quite similar to (1.1.339). The cd f F (x; α, β) has been characterized in Hamedani and Maadooliat (2017).

G. G. Hamedani

90

1.1.169

Odd Log-Logistic Exponentiated Weibull (OLLEW)

The cd f and pd f of OLLEW are given, respectively, by   β γθ −( αx ) 1−e

F (x; α, β, γ, θ) =      θ , x ≥ 0, β γθ β γ −( αx ) −( αx ) + 1− 1−e 1−e

(1.1.341)

and      θ−1 β γθ−1 β γ −( αx ) −( αx ) θβγx e 1−e 1− 1−e f (x; α, β, γ, θ) = , (1.1.342) ( γθ   γ θ )2 x β x β + 1 − 1 − e−( α ) 1 − e−( α ) β

β−1 −( αx )

x > 0, where α, β, γ, θ are all positive parameters. Remark 1.122. Cordeiro et al. (2016) introduced the following distribution (G (x; τ))αθ F (x; α, θ, τ) = h iα , x ∈ R. (G (x; τ))αθ + 1 − (G (x; τ))θ  γ x β Taking G (x; τ)θ = 1 − e−( α ) , x ≥ 0, in F (x; α, θ, τ),we arrive at (1.1.341). The

cd f F (x; α, θ, τ) has been characterized in Hamedani (2019).

1.1.170

Another Odd Log-Logistic Logarithmic (AOLLL-G)

The cd f and pd f of AOLLL-G are given, respectively, by F (x; α, β, φ) = 1−   [log(1 − β)]−1 log 1 −

βG (x; φ)α G (x; φ)α + G (x; φ)α



, x ∈ R,

(1.1.343)

and

f (x; α, β, φ) =

−αβg (x; φ) G (x; φ)α−1 G (x; φ)α−1   , log (1 − β) G (x; φ)α + (1 − β) G (x; φ)α G (x; φ)α + G (x; φ)α

x ∈ R,

(1.1.344)

where α > 0, β ∈ (0, 1) are parameters and G (x; φ) is a baseline cd f with the corresponding pd f g (x; φ), which may depend on the parameter vector φ.

Introduction

1.1.171

91

Type I Half-Logistic (TIHL-G)

The cd f and pd f of TIHL-G are given, respectively, by F (x; λ, η) = and f (x; λ, η) =

1 − [1 − G (x; η)]λ

1 + [1 − G (x; η)]λ

, x ∈ R,

2λg (x; η) [1 − G (x; η)]λ−1 n o2 , x ∈ R, λ 1 + [1 − G (x; η)]

(1.1.345)

(1.1.346)

where λ > 0 is a parameter and G (x; η) is a baseline cd f with the corresponding pd f g (x; η), which may depend on the parameter vector η. Remark 1.123. Dias et al. (2016) introduced the following distribution F (x; α, λ, p) =

(


λ ) α 1 − G (x) ,
λ 1 − p G(x)

x ∈ R,

where α > 0, λ > 0 , p < 1 are parameters and G (x) is a baseline cd f . Taking α = 1, p = −1 in F (x; α, λ, p), we arrive at the cd f (1.1.345). The cd f F (x; α, λ, p) has been characterized in Hamedani and Safavimanesh (2017).

1.1.172

Odd Burr- Generalized (OBu-G)

The cd f and pd f of OBu-G are given, respectively, by F (x; a, b, η) = 1 − and f (x; a, b, η) =

[1 − G (x; η)]ab

{G (x; η)a + [1 − G (x; η)]a }

b

, x ∈ R,

abg (x; η) G (x; η)a−1 [1 − G (x; η)]ab−1 {G (x; η)a + [1 − G (x; η)]a }b+1

, x ∈ R,

(1.1.347)

(1.1.348)

where a > 0, b > 0 are parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η), which may depend on the parameter vector η. Remark 1.124. Cordeiro et al. (2016) introduced the following distribution F (x; α, θ, τ) =

(G (x; τ))αθ h iα , x ∈ R. (G (x; τ))αθ + 1 − (G (x; τ))θ

The characterizations of the cd f (1.1.347) are similar to those of F (x; α, θ, τ), which have been appeared in Hamedani (2019).

G. G. Hamedani

92

1.1.173

Odd Log-Logistic Topp-Leone G (OLLTL-G)

The cd f and pd f of OLLTL-G are given, respectively, by iαθ 1 − G (x; η)2 F (x; α, θ, η) = h iαθ  h iθ α , x ∈ R, 2 2 1 − G (x; η) + 1 − 1 − G (x; η) h

and

f (x; α, θ, η) =

(1.1.349)

h iαθ−1  h iθ α−1 2 2 2αθg (x; η) G (x; η) 1 − G (x; η) 1 − 1 − G (x; η) h

x ∈ R,

1 − G (x; η)

2

iαθ

 h iθ α 2 2 + 1 − 1 − G (x; η)

,

(1.1.350)

where α > 0, θ > 0 are parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η), which may depend on the parameter vector η. Remark 1.125. Cordeiro et al. (2016) introduced the following distribution F (x; α, θ, τ) =

(G (x; τ))αθ h iα , x ∈ R. (G (x; τ))αθ + 1 − (G (x; τ))θ

The characterizations of the cd f (1.1.349) are similar to those of F (x; α, θ, τ), which have been appeared in Hamedani (2019).

1.1.174

Topp-Leone Odd Lindley-G (TLOL-G)

The cd f and pd f of TLOL-G are given, respectively, by F (x; α, λ, φ) =

(

α + G (x; φ) 1− (1 + α) G (x; φ) 

2

G(x;φ) −2α G(x;φ)

e

)λ

, x ∈ R,

(1.1.351)

and

f (x; α, λ, φ) =

  −2α G(x;φ) 2λα2 g (x; φ) α + G (x; φ) e G(x;φ)

× (1 + α)2 G (x; φ)4 ( )λ−1  2 G(x;φ) α + G (x; φ) −2α G(x;φ) 1− e , (1 + α) G (x; φ)

(1.1.352)

x ∈ R, where α > 0, λ > 0,are parameters and G (x; φ) is a baseline cd f with the corresponding pd f g (x; φ), which may depend on the parameter vector φ.

Introduction

93

Remark 1.126. Gomes-Silva et al. (2017) introduced the following distribution   K (x; η) a + K (x; η) F (x; a, η) = 1 − exp −a , x ∈ R. (1 + a) K (x; η) K (x; η) The cd f (1.1.351) is a slightly generalized version of the cd f F (x; a, η). The characterizations of (1.1.351) will be similar to those of F (x; a, η), which have been appeared in Hamedani (2017a).

1.1.175

Odd Log-Logistic Log-Normal (OLL-LN)

The cd f and pd f of OLL-LN are given, respectively, by  α Φ logσx−µ F (x; α, σ, µ) =  α h  iα , x ≥ 0, Φ logσx−µ + 1 − Φ logσx−µ

(1.1.353)

and

f (x; α, σ, µ) =

αφ

  α−1 h  iα−1 Φ logσx−µ 1 − Φ logσx−µ , α h  iα o2 n  xσ Φ logσx−µ + 1 − Φ logσx−µ



log x−µ σ

(1.1.354)

x > 0, where α > 0, σ > 0, µ ∈ R are parameters and Φ (x) and φ (x) are the cd f and pd f of the standard normal distribution. Remark 1.127. Gui (2013) introduced the following distribution h  i p 1 − Φ µ−ln(x) σ i p n h  i p o , F (x; α, µ, σ, p) = h  µ−ln(x) α Φ µ−ln(x) + 1 − Φ σ σ

x ≥ 0.

The cd f (1.1.353) is similar, in some way, to of the cd f F (x; α, µ, σ, p), so the characterizations of (1.1.353) will be similar to those of F (x; α, µ, σ, p), which have appeared in Hamedani (2018b).

1.1.176

Odd Log-Logistic Generalized Gompertz (OLLGG)

The cd f and pd f of OLLGG are given, respectively, by icα b ax 1 − e− a (e −1) F (x; α, a, b, c) = h icα n h iα oα , x ≥ 0, b ax b ax 1 − e− a (e −1) + 1 − 1 − e− a (e −1) h

and

d F (x; α, a, b, c), x > 0, dx where α, a, b, c are all positive parameters. f (x; α, a, b, c) =

(1.1.355)

(1.1.356)

G. G. Hamedani

94

Remark 1.128. Cordeiro et al. (2016) (Remark 1.121) introduced the following distribution (G (x; τ))αθ h iα , x ∈ R. (G (x; τ))αθ + 1 − (G (x; τ))θ h ic b ax Taking G (x; τ) = 1 − e− a (e −1) , x ≥ 0, in F (x; α, θ, τ),we arrive at (1.1.355). The cd f F (x; α, θ, τ) has been characterized in Hamedani (2019). F (x; α, θ, τ) =

1.1.177

New Odd Log-Logistic Half-Logistic (NOLL-HL)

The cd f and pd f of NOLL-HL are given, respectively, by h −x iα F (x; α, β) = h

and

1−e 1+e−x

1−e−x 1+e−x

iα

f (x; α, β) =

h

+ 1−

h

1−e−x 1+e−x

iα iβ , x ≥ 0,

d F (x; α, β), x > 0, dx

(1.1.357)

(1.1.358)

where α > 0, β > 0 are parameters. Remark 1.129. Cordeiro et al. (2016) (Remark 1.121) introduced the following distribution F (x; α, θ, τ) =

(G (x; τ))αθ h iα , x ∈ R. (G (x; τ))αθ + 1 − (G (x; τ))θ

h −x iα Take G (x; τ) = 1−e , x ≥ 0, in F (x; α, θ, τ) and note that characterizations of the 1+e−x cd f (1.1.357) are similar to those of the cd f F (x; α, θ, τ), which has appeared in Hamedani (2019).

1.1.178

Zografos-Balakrishnan Odd Log-Logistic Generalized Half-Normal (ZNOLL-GHN)

The cd f and pd f of ZNOLL-GHN are given, respectively, by

and

1 F (x; α, β, θ, λ) = × Γ (β)   n h
i oα λ   2Φ θx −1  γ β, − log 1 − n h
i oα n h λ   2Φ θx − 1 + 2 − 2Φ

    io , x ≥ 0, α
 x λ 

θ

(1.1.359)

Introduction

95

d F (x; α, β, θ, λ), x > 0, (1.1.360) dx where α, β, θ, λ are all positive parameters and Φ (x) is the cd f of the standard normal. f (x; α, β, θ, λ) =

Remark 1.130. The cd f (1.1.359) was proposed in Altun et al. (2018).

1.1.179

Odd Log-Logistic Marshall-Olkin Generalized Half-Normal (OLLMOGHN)

The cd f and pd f of OLLMOGHN are given, respectively, by

and

oα n h
i λ −1 2Φ θx F (x; α, β, θ, λ) = n h
i oα n h λ 2Φ θx − 1 + β 2 − 2Φ


x λ θ

ioα , x ≥ 0,

(1.1.361)

d F (x; α, β, θ, λ), x > 0, (1.1.362) dx where α, β, θ, λ are all positive parameters and Φ (x) is the cd f of the standard normal. f (x; α, β, θ, λ) =

Remark 1.131. The cd f (1.1.361), as far as the characterizations are concerned, is a bit simpler than cd f (1.1.213).

1.1.180

Power-Linear Hazard Rate (P-LHR)

The cd f and pd f of P-LHR are given, respectively, by b 2 a k+1 + k+1 x

F (x; a, b, k) = 1 − e−( 2 x

), x ≥ 0,

(1.1.363)

and   b 2 a k+1 f (x; a, b, k) = bx + axk e−( 2 x + k+1 x ) , x > 0,

(1.1.364)

where a > 0, b > 0 and k > −1, k 6= 1 are parameters.

Remark 1.132. The cd f (1.1.363) is a special case of the cd f (1.1.17).

1.1.181

Alpha-Power Pareto (APP)

The cd f and pd f of APP are given, respectively, by ( 1−x−β α −1 if α 6= 1 α−1 , F (x; α, β) = , x ≥ 1, −β 1−x , if α = 1 and

(1.1.365)

G. G. Hamedani

96

f (x; α, β) =

(

β log α 1−x−β −β−1 x , α−1 α −β−1

x

,

if α 6= 1 , x > 1, if α = 1

(1.1.366)

where α > 0, β > 0 are parameters.

1.1.182

Exponentiated Odd Log-Logistic Normal (EOLLN)

The cd f and pd f of EOLLN are given, respectively, by F (x; α, β, µ, σ) = and

"

Φ Φ

x−µ
α σ


α x−µ
α  + 1 − Φ x−µ σ σ

#β

, x ∈ R,

(1.1.367)

d F (x; α, β, µ, σ), x ∈ R, (1.1.368) dx where α > 0, β > 0, σ > 0 and µ ∈ R are parameters and Φ (x) is the cd f of the standard normal. f (x; α, β, µ, σ) =

Remark 1.133. da Silva Braga et al. (2016) proposed the following distribution
Φα x−µ σ F (x; α, µ, σ) =

α , x ∈ R, Φα x−µ + 1 − Φ x−µ σ σ

which, as far as characterizations are concerned, is similar to cd f (1.1.367). The cd f F (x; α, µ, σ) has been characterized in Hamedani (2019).

1.1.183

Extended Odd Fréchet-G (EOF-G)

The cd f and pd f of EOF-G are given, respectively, by

and

  −1/β G (x; ψ) F (x; α, β, ψ) = 1 + β , x ∈ R, G (x; ψ)

(1.1.369)

d F (x; α, β, ψ), x ∈ R, (1.1.370) dx where α > 0, β > 0 are parameters and G (x; ψ) is a baseline cd f which may depend on the parameter vector ψ. f (x; α, β, ψ) =

Remark 1.134. Arifa et al. (2017) proposed the following distribution "

G (x; ψ) F (x; α, β, γ, ψ) = 1 + γ G (x; ψ) 

β #− αγ

,

x ∈ R,

which is the same as cd f (1.1.369) for α = 1. The cd f F (x; α, β, γ, ψ) has been characterized in Hamedani (2017).

Introduction

1.1.184

97

Log-Odd Log-Logisticc Birnbaum-Saunders-Poisson (OLLBSP)

The cd f and pd f of OLLBSP are given, respectively, by

and

    βΦ (ν)α 1 exp − 1 , x ∈ R, F (x; α, β) = β e −1 Φ (ν)α + [1 − Φ (ν)]α

(1.1.371)

d F (x; α, β), x ∈ R, (1.1.372) dx where α > 0, β > 0 are parameters , ν = a−1 ρ (x/b), ρ (p) = p1/2 − p−1/2 and Φ (ν) is the standard normal cd f . f (x; α, β) =

Remark 1.135. Characterizations of the cd f (1.1.371) are similar to those of cd f (1.1.213). Please see Remark 1.73 as well.

1.1.185

Zografos-Balakrishnan Lindley-Poisson (ZB-LP)

The cd f and pd f of ZB-LP are given, respectively, by F (x; δ, λ, θ)   = γ δ, − log 1 − x ≥ 0, and

1 1 − eλ

    1 + θ + θx −θx 1 − exp λ 1 − e , 1+θ

(1.1.373)

λθ2 e−θx

   1 + θ + θx −θx f (x; δ, λ, θ) = e ×
exp λ 1 − 1+θ (1 + θ)2 1 − eλ δ−1      1 1 + θ + θx −θx − log 1 − 1 − exp λ 1 − e , 1+θ 1 − eλ

(1.1.374)

x > 0, where δ, λ, θ are parameters γ (α, x) is the incomplete gamma function.

1.1.186

One-Parameter Weibull-Type (1P-Weibull)

The cd f and pd f of 1P-Weibull are given, respectively, by x

F (x; w) = 1 + w



 2 + log (w) [x (x + 1) log(w) − 2x − 1] , x ≥ 0, log(w) − 2

(1.1.375)

log3 (w) x (x + 1) wx , x > 0, log(w) − 2

(1.1.376)

and f (x; w) = where w ∈ (0, 1) is a parameter.

G. G. Hamedani

98

1.1.187

Beta Odd Lindley-G (BOL-G)

The cd f and pd f of BOL-G are given, respectively, by F (x; a, b, λ) =

1 × B (a, b)

Z 1− λ+G(x) exp −λ G(x) (1+λ)G(x)α G(x) 0

n

o

t a−1 (1 − t)b−1 dt, x ∈ R,

(1.1.377)

and   a−1 1 λ + G (x) G (x) f (x; a, b, λ) = 1− exp −λ B (a, b) (1 + λ) G (x) G (x)   b−1 λ + G (x) G (x) × exp −λ × (1 + λ) G (x) G (x)   G (x) λ2 g (x) exp −λ , G (x) (1 + λ) G (x)3

(1.1.378)

x ∈ R, where a, b, λ are all positive parameters and G (x) is a baseline cd f with pd f g (x).

1.1.188

A Distribution For Instantaneous Failures (ADFIF)

The cd f and pd f of ADFIF are given, respectively, by
λ2 + x − λ −x/λ F (x; λ) = 1 − e , x ≥ 0, λ (λ − 1) and
λ2 + x − 2λ −x/λ f (x; λ) = e , x > 0, λ2 (λ − 1)

(1.1.379)

(1.1.380)

where λ ≥ 2 is a parameter.

1.1.189

Generalized Inverse Weibull- Generalized Inverse Weibull (GIWGIW)

The cd f and pd f of GIW-GIW are given, respectively, by "   # α  x b β F (x; α, β, θ, a, b, c) = 1 − exp −θ , x ≥ 0, c a

(1.1.381)

and

"     # θβαb  x b−1 α  x b β−1 α  x b β f (x; α, β, θ, a, b, c) = exp −θ , x > 0, ac a c a c a (1.1.382)

Introduction

99

where α, β, θ, a, b, c are all positive parameters. Remark 1.136. The cd f (1.1.381) can be expressed as follows which is not new.    α β bβ x , x ≥ 0. F (x; α, β, θ, a, b, c) = 1 − exp −θ cab

1.1.190

Beta Lindley Geometric (BLGc)

The cd f and pd f of BLGc are given, respectively, by

1 F (x; θ, p, a, b) = B (a, b)

Z

( (

) )

1− 1+ θθx e−θx θ+1 θθx e−θx 1−p 1+ θ+1

0

t a−1 (1 − t)b−1 dt , x ≥ 0,

(1.1.383)

and d F (x; θ, p, a, b), x > 0, dx where θ, a, b all positive and p ∈ (0, 1) are parameters. f (x; θ, p, a, b) =

Remark 1.137. Taking Q (x) = (1.1.383).

1.1.191

θθx 1−(1+ θ+1 )e−θx θθx 1−p(1+ θ+1 )e−θx

(1.1.384)

, x ≥ 0, in Remark 1.15, we arrive at the cd f

Odd Log-Logistic Generalized Half-Normal (OLLGHN)

The cd f and pd f of OLLGHN are given, respectively, by

and

n h
i oα λ 2Φ θx −1 F (x; α, β, θ, λ) = n h
i oα n h λ 2Φ θx − 1 + 2 − 2Φ


x λ θ

ioβ , x ≥ 0,

d F (x; (x; α, β, θ, λ)) , x > 0, dx where α, β, θ, λ are all positive parameters. f (x; α, β, θ, λ) =

(1.1.385)

(1.1.386)

Remark 1.138. The cd f (1.1.361) is n h
i oα λ 2Φ θx −1 F (x; α, β, θ, λ) = n h
i oα n h λ 2Φ θx − 1 + β 2 − 2Φ


x λ θ

ioα ,

which is very similar to the cd f (1.1.385), so similar characterizations can be stated for the OLLGHN distribution.

G. G. Hamedani

100

1.1.192

Odd Lindley Lomax (OLLo)

The cd f and pd f of OLLo are given, respectively, by nh i h io F (x; β, λ, a) = 1 − a + (1 + x/β)−λ / (1 + a) (1 + x/β)−λ × n h io exp −a (1 + x/β)λ − 1 , (1.1.387)

and

f (x; β, λ, a) =

d F (x; β, λ, a), x > 0, dx

(1.1.388)

where β, λ, a are all positive parameters. Remark 1.139. The cd f (1.1.387) is similar to the cd f (1.1.87), so similar characterizations can be stated for the OLLo distribution.

1.1.193

Burr X Exponentiated Lomax (BrXELx)

The cd f and pd f of BrXELx are given, respectively, by   h 2 θ
iα      1 − xβ−1 + 1 −λ    F (x; α, β, θ, λ) = 1 − exp − h iα  , −λ     1 − 1 − (xβ−1 + 1) 

(1.1.389)

x ≥ 0, and

d F (x; α, β, θ, λ), x > 0, (1.1.390) dx where α, β, θ, λ are all positive parameters. h
−λ iα Remark 1.140. Taking G (x) = 1 − xβ−1 + 1 , x ≥ 0, in the cd f (1.1.7), we arrive at the cd f (1.1.389). f (x; α, β, θ, λ) =

1.1.194

Extended Normal (EN)

The cd f and pd f of EN are given, respectively, by

and

   a b x−µ F (x; µ, σ, a, b) = 1 − 1 − Φ , x ∈ R, σ    a−1 ab x−µ x−µ f (x; µ, σ, a, b) = φ 1−Φ σ σ σ    a b−1 x−µ × 1− 1−Φ , x ∈ R, σ

(1.1.391)

(1.1.392)

Introduction

101

where µ ∈ R, σ > 0, a > 0 and b > 0 are parameters and Φ (x) , φ (x) are cd f and pd f of the standard normal.
, in the cd f (1.1.391), we Remark 1.141. Taking G (x) = Φ x−µ σ b

F (x; a, b) = [1 − {1 − G (x)}a ] , x ∈ R,

which has been characterized in Hamedani(2016). Characterizations of the cd f (1.1.391) will be similar to those of F (x; a, b).

1.1.195

Poisson-X (P-X)

The cd f and pd f of P-X are given, respectively, by F (x; c, η) = and f (x; c, η) =

i 1 h −(G(x;η))c 1 − e , 1 − e−1

x ∈ R,

h i c c−1 −(G(x;η))c g (x; η) (G (x; η)) 1 − e , x ∈ R, 1 − e−1

(1.1.393)

(1.1.394)

where c > 0 is a parameter and G (x; η) , g (x; η) are cd f and pd f of a baseline distribution. Remark 1.142. Nadarajah et al. (2009) proposed the following distribution F (x) =

1 − e−λG1 (x) , x ∈ R, 1 − e−λ

where G1 (x) is a baseline cd f . Taking G1 (x) = (G (x; η))c , x ∈ R and λ = 1, in F (x), we arrive at cd f (1.1.393). The characterizations of the cd f F (x) have appeared in Hamedani (2016).

1.1.196

α-Power Transmuted Generalized Exponential (αPTGE)

The cd f and pd f of αPTGE are given, respectively, by  β  α(1−e−λx ) −1 , if α 6= 1 , α−1 F (x; α, β, λ) =
β  −λx 1−e , if α = 1

x ≥ 0,

(1.1.395)

if α 6= 1 ,

(1.1.396)

and

f (x; α, β, λ) =

 β  log(α)βλe−λx (1−e−λx)β−1 α(1−e−λx ) −1 

e−λx


α−1 β−1 −λx 1−e ,

x > 0, where α, β are all positive parameters.

,

if α = 1

G. G. Hamedani

102 Remark 1.143. The cd f

F (x; α) =

 αG(x)−1 α−1

G(x)

, i f α 6= 1 , if α = 1

, x ∈ R,

is well known and has already been characterized in our previous work. Taking G (x) =
β 1 − e−λx , x ≥ 0, we arrive at the cd f (1.1.395).

1.1.197

Weibull Exponentiated Exponential (WEE)

The cd f and pd f of WEE are given, respectively, by h n oic α − − log 1−(1−e−λx )

F (x; c, α, λ) = 1 − e

,

x ≥ 0,

(1.1.397)

and
α−1 h n  α oic−1 cαλe−λx 1 − e−λx f (x; c, α, λ) = − log 1 − 1 − e−λx ×
α 1 − 1 − e−λx h n oic α − − log 1−(1−e−λx)

e

,

(1.1.398)

x > 0, where c, α, λ are all positive parameters. Remark 1.144. Ghosh and Nadarajah (2018), proposed the following distribution   1 c F (x; γ, c) = 1 − exp − c [− log(1 − G (x))] , x ∈ R, γ
α where G (x) is a baseline cd f . Taking G (x) = 1 − e−λx , x ≥ 0, γ = 1, in F (x; γ, c), we arrive at the cd f (1.1.397). Thus, WL is a special case of F (x; γ, c). Please see Remark 1.60 for further details.

1.1.198

Weighted Modified Weibull (WMW)

The cd f and pd f of WMW are given, respectively, by −λx−λxk

and

α1−e −1 F (x; α, λ, k) = , α−1 f (x; α, λ, k) =

x ≥ 0,

 λ log(α)  k −λx−λxk 1 + kxk−1 e−λx−λx α1−e , α−1

x > 0, where α, λ, k are all positive parameters.

(1.1.399)

(1.1.400)

Introduction

103

Remark 1.145. The cd f F (x; α) =

 αG(x)−1 α−1

G(x)

, i f α 6= 1 , if α = 1

, x ∈ R,

is well known and has already been characterized in our previous work. Taking G (x) = k 1 − e−λx−λx , x ≥ 0, we arrive at the cd f (1.1.399).

1.1.199

Poisson Nadarajah-Haghighi (PNH)

The cd f and pd f of PNH are given, respectively, by  n oβ  1−(1+λx)α 1 − exp − 1 − e F (x; α, β, λ) = , 1 − e−1 and

f (x; α, β, λ) =

βαλ (1 + λx)α−1 e1−(1+λx)

α

n

1 −e−1  n o α β exp − 1 − e1−(1+λx) ,

x ≥ 0,

α

1 − e1−(1+λx)

(1.1.401)

oβ−1

× (1.1.402)

x > 0, where α, β, λ are all positive parameters. Remark 1.146. Nadarajah et al. (2009) proposed the following distribution 1 − e−λG(x) , x ∈ R, 1 − e−λ n o α β where G (x) is a baseline cd f . Taking G (x) = 1 − e1−(1+λx) , x ≥ 0, λ = 1, in F (x), we arrive at cd f (1.1.401). The cd f F (x) has been characterized in Hamedani (2016). F (x) =

1.1.200

Poisson Burr X Weibull (PBrXW)

The cd f and pd f of PBrXW are given, respectively, by

F (x; α, β, θ, λ) =

(

)!   2 θ β 1 − exp −λ 1 − exp − e(αx) − 1 1 − e−λ

, x ≥ 0,

(1.1.403)

and d F (x; α, β, θ, λ), dx where α, β, θ, λ are all positive parameters. f (x; α, β, θ, λ) =

x > 0,

(1.1.404)

G. G. Hamedani

104

Remark 1.147. Nadarajah et al. (2009) proposed the following distribution F (x) =

1 − e−λG(x) , x ∈ R, 1 − e−λ ( 

where G (x) is a baseline cd f . Taking G (x) =

1 − exp −



β e(αx)

−1

2 θ

)

, x ≥ 0, in

F (x), we arrive at cd f (1.1.403). The cd f F (x) has been characterized in Hamedani (2016).

1.1.201

Marshall-Olkin Extended Exponential (MOEE)

The cd f and pd f of MOEE are given, respectively, by
α 1 − e−βx F (x; α, β, λ) =
α , x ≥ 0, λ + λ 1 − e−βx

(1.1.405)

and


α−1 αβλe−βx 1 − e−βx f (x; α, β, λ) = h
α i2 , −βx λ+λ 1−e

x > 0,

(1.1.406)

where α, β, λ are all positive parameters.

Remark 1.148. Hossain et al. (2019) proposed a distribution with cd f (1.1.241) F (x; γ, b, k) = Taking k = b = 1 , γ =

1.1.202

1 λ

"

γG (x)b

#k

, x ∈ R. 1 − (1 − γ) G (x)b
α and G (x) = 1 − e−βx , x ≥ 0, we arrive at the cd f (1.1.405).

New Generalization of Weibull-Exponential (NGWE)

The cd f and pd f of NGWE are given, respectively, by

and

h  a ib λx F (x; α, λ, a, b) = 1 − 1 − 1 − e−α(e −1) , x ≥ 0,

d F (x; α, λ, a, b), x > 0, dx where α, λ, a, b are all positive parameters. f (x; α, λ, a, b) =

(1.1.407)

(1.1.408)

Remark 1.149. El-Damcese et al. (2016) introduced the following distribution ioiθ h n h  d F (x; a, b, c, d, θ) = 1 − exp − axb ecx − 1 , x ≥ 0.

Characterizations of the cd f (1.1.407) are similar to those of the cd f F (x; a, b, c, d, θ), which have appeared in Hamedani and Najibi (2016).

Introduction

1.1.203

105

Odd Fréchet Inverse Weibull (OFIW)

The cd f and pd f of OFIW are given, respectively, by  h iθ  x−α/β F (x; α, β, θ) = exp − e −1 , x ≥ 0,

(1.1.409)

and

 h iθ−1 iθ  αβθ x−α/β h x−α/β x−α/β f (x; α, β, θ) = β+1 e e −1 exp − e −1 , x > 0, x

(1.1.410)

where α, β, θ are all positive parameters. Remark 1.150. The cd f (1.1.181) has the following form     β x−β β x−β θ −θbδ −bδ 1 − exp −ae 2−e , x ≥ 0. F (x; β, θ, δ, a, b) = 1 − e−a Clearly, characterizations similar to those of the cd f (1.1.181) can be stated for the cd f (1.1.409).

1.1.204

Topp-Leone-G Poisson (TL-GP)

The cd f and pd f of TL-GP are given, respectively, by "

1 − e−λG1 (x;ψ) F (x; α, λ, ψ) = 1 − e−λ

#α "

1 − e−λG1 (x;ψ) 2− 1 − e−λ

#α

, x ∈ R,

(1.1.411)

and d F (x; α, λ, ψ), x ∈ R, (1.1.412) dx where α, λ, ψ are all positive parameters and G1 (x; ψ) is a baseline cd f which may depend on the parameter vector ψ. f (x; α, λ, ψ) =

Remark 1.151. Taking G (x) = cd f (1.1.411).

1.1.205

1−e−λG1 (x;ψ) , 1−e−λ

x ∈ R, in the cd f (1.1.307), we arrive at the

Alpha Power Transmuted Extended Exponential (APTEE)

The cd f and pd f of APTEE are given, respectively, by  γ+β−(γ+β+γβx)e−γx α γ+β −1 , if α 6= 1 , x ≥ 0 α−1 F (x; α, β, γ) = −γx  γ+β−(γ+β+γβx)e , if α = 1 γ+β

and

(1.1.413)

G. G. Hamedani

106

f (x; α, β, γ) =

d F (x; α, β, γ), x > 0, dx

(1.1.414)

where α, β, γ are all positive parameters. Remark 1.152. Taking G (x) =

γ+β−(γ+β+γβx)e−γx γ+β

F (x; α, β, γ) =

, x ≥ 0, the cd f (1.1.413) can be expressed as ( G(x) ) α −1 α−1 , i f G(x) , if

α 6= 1 α =1

,

which is well-known and has been characterized in our previous work.

1.1.206

Odd Fréchet Inverse Rayleigh (OFIR)

The cd f and pd f of OFIR are given, respectively, by  h iθ  x−α/2 F (x; α, θ) = exp − e −1 , x ≥ 0,

(1.1.415)

and

iθ−1 2αθ x−α/2 h x−α/2 e e − 1 × x3   h −α/2 iθ exp − ex −1 , x > 0,

f (x; α, θ) =

(1.1.416)

where α, θ are all positive parameters. Remark 1.153. For β = 2, the cd f (1.1.409) reduces to the cd f (1.1.415).

1.1.207

Odd Inverse Pareto-G (OIP-G)

The cd f and pd f of OIP-G are given, respectively, by  −α G (x; ϕ)α G (x; ϕ) β + G (x; ϕ)α G (x; ϕ)  α G (x; ϕ) = , x ∈ R, β + (1 − β) G (x; ϕ)

F (x; α, β, ϕ) =

(1.1.417)

and d F (x; α, β, ϕ), x ∈ R, (1.1.418) dx where α, β are all positive parameters and G (x; ϕ) is a baseline cd f which may depend on the parameter vector ϕ. f (x; α, β, ϕ) =

Remark 1.154. For γ = β1 , the cd f (1.1.241) reduces to the cd f (1.1.417).

Introduction

1.1.208

107

Type I Half-Logistic Burr X (TIHLBX )

The cd f and pd f of TIHLBX are given, respectively, by

F (x; α, θ, λ) =

and

   λ 2 θ 1 − 1 − 1 − e−(αx)

   λ , x ≥ 0, 2 θ 1 + 1 − 1 − e−(αx)

(1.1.419)

d F (x; α, θ, λ), x > 0, (1.1.420) dx where α, θ, λ are all positive parameters.   2 θ Remark 1.155. Taking G (x; α, θ) = 1 − e−(αx) , x ≥ 0, the cd f (1.1.419) can be expressed as f (x; α, θ, λ) =

 λ 1 − G (x; α, θ) F (x; α, θ, λ) =  λ , x ≥ 0. 1 + G (x; α, θ)

Dias et al. (2016) introduced the following distribution F (x; α, λ, p) =

(


λ ) α 1 − G (x) ,
λ 1 − p G(x)

x ∈ R,

where α > 0, λ > 0 , p < 1 are parameters and G (x) is a baseline cd f . Taking p = −1, the cd f F (x; α, λ, p) provides a more general form of that of (1.1.419). The cd f F (x; α, λ, p) has been characterized in Hamedani and Safavimanesh (2017).

1.1.209

Geometric Lindley Poisson 1 (GLP1)

The cd f and pd f of GLP1 are given, respectively, by 
−θx
 (1 − π) 1 − exp −λ + λ θ+1+θx e 1+θ 

, F (x; θ, λ, π) = θ+1+θx 1 − e−λ − π 1 − exp −λ + λ 1+θ e−θx

(1.1.421)

and

d F (x; θ, λ, π), x > 0, (1.1.422) dx where θ > 0, λ > 0 and π ∈ (0, 1) are parameters.
−θx Remark 1.156. Taking G (x; θ) = 1 − θ+1+θx e , x ≥ 0, the cd f (1.1.421) can be ex1+θ pressed as f (x; θ, λ, π) =

G. G. Hamedani

108

(1 − π) [1 − exp (−λG (x; θ))] 1 − e−λ − π [1 − exp (−λG (x; θ))] h i 1−exp(−λG(x;θ)) (1 − π) −λ 1−e h i = 1−exp(−λG(x;θ)) 1−π 1−e−λ h i

F (x; θ, λ, π) =

=

Now taking β = written as

1 1−π

1−exp(−λG(x;θ)) 1−e−λ

1 1−π

and K (x; θ) =

F (x; θ, λ, π) =

π − 1−π

h

h

1−exp(−λG(x;θ)) 1−e−λ

1−exp(−λG(x;θ)) 1−e−λ

i

i.

, x ≥ 0, the cd f (1.1.421) can be

K (x; θ) , x ≥ 0, β + (1 − β) K (x; θ)

which is a special case of the cd f (1.1.417).

1.1.210

Odd Fréchet Inverse Exponential (OFIE)

The cd f and pd f of OFIE are given, respectively, by  h iθ  αx−1 F (x; α, θ) = exp − e −1 , x ≥ 0,

(1.1.423)

and

 h iθ−1 iθ  αθ αx−1 h αx−1 αx−1 f (x; α, θ) = 3 e e −1 exp − e −1 , x > 0, x

(1.1.424)

where α, θ are all positive parameters. Remark 1.157. The cd f (1.1.423) is a special case of the cd f (1.1.415).

1.1.211

Truncated Weibull Power Lomax (TWPL)

The cd f and pd f of TWPL are given, respectively, by     β

− 1− 1+ xγ

and

F (x; α, β, γ, λ) = A 1 − e

−α

λ 

 , x ≥ 0,

d F (x; α, β, γ, λ), x > 0, dx
−1 where α, β, γ, λ are all positive parameters and A = 1 − e−1 . f (x; α, β, γ, λ) =

(1.1.425)

(1.1.426)

Introduction Remark 1.158. The G (x; α, β, γ, λ) = have the following form





1− 1+

xβ γ

109 −α λ

, x ≥ 0, the cd f (1.1.425) will

  F (x; α, β, γ, λ) = A 1 − e−G(x;α,β,γ,λ) , x ≥ 0,

which has been characterized in our previous work.

1.1.212

Generalized Extended Inverse Weibull (GEIW)

The cd f and pd f of GEIW are given, respectively, by   θ −β −λx F (x; α, β, θ, λ) = 1 − 1 − exp −αx e , x ≥ 0,

and

(1.1.427)

d F (x; α, β, θ, λ), x > 0, (1.1.428) dx where α, β, θ, λ are all positive parameters.
Remark 1.159. Taking α = c = 1, G (x) = exp −αx−β e−λx , x ≥ 0 , in the cd f (1.1.203), we arrive at the cd f (1.1.427). f (x; α, β, θ, λ) =

1.1.213

Lomax-Lindley (L-L)

The cd f and pd f of L-L are given, respectively, by "

    −α # 1 θx −θx F (x; α, β, θ) = K 1 − 1 + 1− 1+ e , x ≥ 0, β θ+1

(1.1.429)

and Kαθ2 (x + 1) e−θx × β (θ + 1)     −α−1 1 θx 1+ 1− 1+ e−θx , x > 0, β θ+1

f (x; α, β, θ) =

(1.1.430)

where α, β, θ are all positive parameters.

1.1.214

Generalized Gompertz-Generalized Gompertz (GG-GG)

The cd f and pd f of GG-GG are given, respectively, by 

− ab

 F (x; α, β, λ, a, b, c) = 1 − 1 − e



(

)

− α eβx −1 1−e β

−bλ ! c

  , x ≥ 0,

(1.1.431)

G. G. Hamedani

110 and

d F (x; α, β, λ, a, b, c), x > 0, (1.1.432) dx where α, β, λ, a, b, c are all positive parameters. h ibλ − α eβx −1) Remark 1.160. Taking G (x; α, β, λ, b) = 1 − e β ( , x ≥ 0, the cd f (1.1.431) can be expressed as  h i c G(x;α,β,λ,b) − ab G(x;α,β,λ,b) F (x; α, β, λ, a, b, c) = 1 − 1 − e , x ≥ 0. f (x; α, β, λ, a, b, c) =

Alizadeh et al. (2016) proposed the following distribution (  θ ) K (x; γ) F (x; α, γ) = exp − , x ∈ R, K (x; γ)

where K (x; θ) is a baseline cd f . The characterizations of cd f (1.1.431) will be similar to those of the cd f F(x; α, γ) which appeared in Hamedani and Maadooliat (2017).

1.1.215

Exponentiated Weibull Weibull (EWW)

The cd f and pd f of EWW are given, respectively, by  a β λxγ F (x; α, β, λ, a) = 1 − e−α(e −1) , x ≥ 0,

(1.1.433)

and

d F (x; α, β, λ, a), x > 0, dx where α, β, λ, a are all positive parameters. f (x; α, β, λ, a) =

(1.1.434)

Remark 1.161. For γ = 2, the cd f (1.1.433) reduces to the cd f proposed by Elgarhy et al. (2019). Characterizations of the cd f (1.1.433) will be the same as for the case γ = 2, which are included in Elgarhy et al. (2019).

1.1.216

Weibull-Inverse Lomax (WIL)

The cd f and pd f of WIL are given, respectively, by (   )  γ λ −β β , x ≥ 0, F (x; α, β, λ, γ) = 1 − exp −α 1 − 1 + x

and

(1.1.435)

Introduction

111

γ λ−1 αβ λγβ  1 + × f (x; α, β, λ, γ) = x2 x ( )      λ −β γ λ γ 1− 1+ exp αβ 1 − 1 + , x > 0, x x

(1.1.436)

where α, β are positive , λ,a scale and γ a location parameters. Remarks 1.162. (a) We believe that the formulas for the cd f and pd f given in Falgore et al. paper are incorrect. The formulas given in (1.1.435) and (1.1.436) are the correct ones. (b) The characterizations of the cd f (1.1.435) are similar to those of the cd f (1.1.59).

1.1.217

Burr XII Inverse Rayleigh (BXII-IR)

The cd f and pd f of BXII-IR are given, respectively, by 2

(

e−(a/x)

F (x; α, β, a) = 1 − 1 +

2

1 − e−(a/x)

and f (x; α, β, a) =

!)−β

d F (x; α, β, a), dx

, x ≥ 0,

(1.1.437)

x > 0,

(1.1.438)

where α, β, a are positive parameters. Remark 1.163. Alizadeh et al. (2019) proposed the following distribution G (x; η) F (x; α, β, η) = 1 − 1 + β G (x; η) 



α − β1

x ∈ R.

,

2

Taking G (x; η) = e−(a/x) , x ≥ 0 in he cd f F (x; α, β, η), we arrive at 2

(

F (x; α, β, η) = 1 − 1 + β

!)− 1

β

e−(a/x)

2

1 − e−(a/x)

, x ≥ 0.

The characterizations of the cd f (1.1.437) are similar to those of F (x; α, β, η),which have appeared in Hamedani (2019).

1.1.218

Kumarswamy Type I Half Logistic (KwTIHL-G)

The cd f and pd f of KwTIHL-G are given, respectively, by (

F (x; a, b, λ) = 1 − 1 − and

"

1 − G (x)λ 1 + G (x)λ

# a )b

, x ∈ R,

(1.1.439)

G. G. Hamedani

112

h ia−1 2abλg (x) G (x)λ−1 1 − G (x)λ f (x; a, b, λ) = × ia+1 h λ 1 + G (x) ( " #a )b−1 1 − G (x)λ 1− , x ∈ R, 1 + G (x)λ

(1.1.440)

where a, b, λ are all positive parameters and G (x) is a baseline cd f with the corresponding pd f g (x).

1.1.219

Odd Generalized Exponential Power Function (OGEPF)

The cd f and pd f of OGEPF are given, respectively, by

and

  α  θ −λ γαx−xα , 0 ≤ x ≤ γ, F (x; α, γ, θ, λ) = 1 − e

α

α

(1.1.441)

 α  x α −2 −λ γα −xα

f (x; α, γ, θ, λ) = αθλγ (γ − x ) e ×   α  θ−1 −λ γαx−xα 1−e , x > 0,

(1.1.442)

where α, γ, θ, λ are all positive parameters.  α Remark 1.164. Taking G (x) = xγ , 0 ≤ x ≤ γ, the cd f (1.1.441) can be expressed as   θ  G(x) −λ G(x) , F (x; α, γ, θ, λ) = 1 − e

which is a special case of the cd f (1.1.7).

1.1.220

Generalized Odd Lomax Generated (GOLG)

The cd f and pd f of GOLG are given, respectively, by 

and

G (x; ψ)θ

−α

 F(x; α, β, θ, ψ) = 1 − 1 +  θ β 1 − G (x; ψ)

, x ∈ R,

(1.1.443)

d F(x; α, β, θ, ψ), x ∈ R, (1.1.444) dx where α, β, θ are all positive parameters and G(x; ψ) is a baseline cd f which may depend on the parameter vector ψ. f (x; α, β, θ, ψ) =

Remark 1.165. Taking K (x) = G(x; ψ)θ , x ∈ R, the cd f (1.1.443) becomes cd f (1.1.57).

Introduction

1.1.221

113

Inverse Weibull Geometric (IWG)

The cd f and pd f of IWG are given, respectively, by F(x; α, λ, p) = e−λx

−α

and

  −1 −α 1 − (1 − p) 1 − e−λx , x ≥ 0,

f (x; α, λ, p) = pαλx−(α+1)e−λx

−α

  −2 −α 1 − (1 − p) 1 − e−λx , x > 0,

(1.1.445)

(1.1.446)

where α > 0, λ > 0 and p ∈ (0, 1) are parameters. −α

Remark 1.166. Taking G (x) = e−λx , x ≥ 0, the cd f (1.1.445) becomes the cd f given in Remark 1.101.

1.1.222

Inverse Weibull Poisson (IWP)

The cd f and pd f of IWP are given, respectively, by   −λx−α F(x; α, λ, β) = (1 − e−β )−1 1 − e−βe , x ≥ 0,

(1.1.447)

and

−α

f (x; α, λ, β) = βαλx−(α+1)e−λx e(1−βe

−λx−α )

, x > 0,

(1.1.448)

where α, λ, β are all positive parameters. −α

Remark 1.167. Taking G (x) = e−λx , x ≥ 0, the cd f (1.1.447) becomes the cd f given in Remark 1.30.

1.1.223

Transmuted Four Parameters Generalized Log-Logistic (TFPGLL)

The cd f and pd f of TFPGLL are given, respectively, by

F(x; α, β, θ, λ) = and

α + xβ


θ n


θ

o − (1 − λ) αθ − λα2θ , x ≥ 0,
2θ α + xβ

α + xβ

n o
θ βθαθ xβ−1 (1 − λ) α + xβ + 2λαθ f (x; α, β, θ, λ) = , x>0
2θ+1 α + xβ

(1.1.449)

(1.1.450)

where α, β, θ all positive and |λ| ≤ 1 are parameters.   α Remark 1.168. Taking G (x) = 1 − α+x , x ≥ 0, the cd f (1.1.449) becomes the cd f β given in Remark 1.27.

G. G. Hamedani

114

1.1.224

Burr X-Kumaraswamy (BXKw)

The cd f and pd f of BXKw are given, respectively, by   F(x; θ, c, d) = 1 − exp −  

and

!2 θ  1 − (1 − x )  , 0 ≤ x ≤ 1,  (1 − xc )d c d

d F(x; θ, c, d), 0 < x < 1, dx where θ, c, d are all positive parameters. f (x; θ, c, d) =

(1.1.451)

(1.1.452)

Remark 1.169. Taking G (x) = 1 − (1 − xc )d , 0 ≤ x ≤ 1, the cd f (1.1.451) can be expressed as   θ  G(x) 2 − G(x) , F (x; θ, c, d) = 1 − e

which is a special case of the cd f (1.1.7).

1.1.225

Transmuted Generalized Gamma (TGG)

The cd f and pd f of TGG are given, respectively, by "

x ≥ 0, and

η k

+ 1, xk λ−k
F(x; α, η, k) = (α + 1) 1 − Γ ηk + 1 "
#2 Γ ηk + 1, xk λ−k
−α 1− , Γ ηk + 1 f (x; α, η, k) =

Γ


#

d F(x; α, η, k), x > 0, dx

(1.1.453)

(1.1.454)

where |α| ≤ 1, η > 0, k > 0 are parameters and Γ (a, x) = x∞ t a−1e−t dt.   Γ( ηk +1,xk λ−k ) Remark 1.170. Taking G (x) = 1 − , x ≥ 0, the cd f (1.1.453) can be exΓ( ηk +1) pressed as R

F(x; α, η, k) = (α + 1) G (x) − αG (x)2 , which is proposed by Bourguignon et al. (2016). Please see Remark 1.27.

Introduction

1.1.226

115

Generalized Extended Exponential-Weibull (GEEW)

The cd f and pd f of GEEW are given, respectively, by

and

   α F(x; α, β, γ, λ, c) = 1 − exp − (βxγ + λx)c , x ≥ 0,

d F(x; α, β, γ, λ, c), x > 0, dx where α, β, λ, c all positive and γ ∈ (0, ∞)\ {1} are parameters. f (x; α, β, γ, λ, c) =

(1.1.455)

(1.1.456)

Remark 1.171. The cd f (1.1.433) is more general than the cd f (1.1.455).

1.1.227

Nadarajah-Haghigh Geometric (NHG)

The cd f and pd f of NHG are given, respectively, by   1 − exp 1 − (1 + λx)α   , x ≥ 0, F(x; α, λ, p) = 1 − p exp 1 − (1 + λx)α and

(1.1.457)

d F(x; α, λ, p), x > 0, (1.1.458) dx where α > 0, λ > 0, p ∈ (0, 1) are parameters.   Remark 1.172. Taking G (x) = 1 − exp 1 − (1 + λx)α , x ≥ 0, the cd f (1.1.457) can be written as f (x; α, λ, p) =

F(x; α, λ, p) = which is mentioned in Remark 1.53.

1.1.228

G (x) , 1 − pG (x)

Nadarajah-Haghighi Lindley (NHL)

The cd f and pd f of NHL are given, respectively, by F(x; α, λ, γ) = 1 −

(1 + γ + γx) 1−γx−(1+λx)α e , x ≥ 0, 1+γ

(1.1.459)

and d F(x; α, λ, γ), x > 0, dx where α ≥ 0, λ ≥ 0, γ ≥ 0, with max(α, γ) > 0, max(λ, γ) are parameters. f (x; α, λ, γ) =

(1.1.460)

Remark 1.173. Ghitany et al. (2013) introduced the following distribution   1 + β + βxα F (x; α, β) = 1 − exp [−βxα ], x ≥ 0, 1+β which is the cd f (1.1.85). Similar characterizations can be stated for the cd f (1.1.459).

G. G. Hamedani

116

1.1.229

Exponentiated Log-Sinh Cauchy (ELSC)

The cd f and pd f of ELSC are given, respectively, by τ  1 1 + arctan [ν sinh(w)] , x ≥ 0, F(x; µ, σ, ν, τ, p) = (1 − p) 2 π and f (x; µ, σ, ν, τ, p) =

d F(x; µ, σ, ν, τ), x > 0, dx

where µ ∈ R, σ > 0, ν > 0, τ > 0, p ∈ (0, 1) are parameters and w =

(1.1.461)

(1.1.462) log(x)−µ . σ

Remark 1.174. Characterizations similar to those of cd f (1.1.205) can be stated for the cd f (1.1.461).

1.1.230

Modified Fréchet (MFr)

The cd f and pd f of MFr are given, respectively, by     α β −λx F(x; α, β, λ) = exp − e , x ≥ 0, x

(1.1.463)

and

d F(x; α, β, λ), x > 0, dx where α > 0, β > 0, λ ≥ 0 are parameters. f (x; α, β, λ) =

1.1.231

(1.1.464)

Cubic Transmuted Uniform (CTU)

The cd f and pd f of CTU are given, respectively, by F(x; λ) = (1 − λ) G (x) + 3λG (x)2 − 2λG (x)3 , x ∈ R,

(1.1.465)

n o f (x; λ) = g (x) 1 − λ + 6λG (x) − 6λG (x)2 , x ∈ R,

(1.1.466)

and

where |λ| ≤ 1 is a parameter and G (x) is a baseline cd f .

1.1.232

Beta-G Poisson (BGP)

The cd f and pd f of BGP are given, respectively, by F(x; a, b, ϕ, λ) =

1 − exp [−λH (x; a, b, ϕ)] , x ∈ R, 1 − exp (−λ)

(1.1.467)

d F(x; a, b, ϕ, λ), x ∈ R, dx

(1.1.468)

and f (x; a, b, ϕ, λ) =

Introduction

all positive parameters and H (x; a, b, ϕ) = is a baseline cd f which may depend on the parameter vector ϕ and G (x; ϕ) is a cd f . where

a, b, λ

117

are

1 R G(x;ϕ) a−1 w (1 − w)b−1 dw B(a,b) 0

Remark 1.175. Nadarajah et al. (2009) proposed the following distribution 1 − e−λG1 (x) , x ∈ R, 1 − e−λ G1 (x) is a baseline cd f . Taking G1 (x) = H (x; a, b, ϕ) in the above F (x) we arrive at the cd f (1.1.467). The characterizations of the cd f (1.1.467) are similar to those of F (x) which have appeared in Hamedani (2016). F (x) =

1.1.233

Cosine-Sine Transformation (CST)

The cd f and pd f of CST are given, respectively, by
(α + γ) sin π2 G (x)

F(x; α, β, γ, θ) = , α + β cos π2 G (x) + γ sin π2 G (x) + θ2 sin(πG (x))

(1.1.469)

x ∈ R, and



3  g (x) π (α + γ) β + α cos π2 G (x) + θ sin π2 G (x) f (x; α, β, γ, θ) = 

2 , α + β cos π2 G (x) + γ sin π2 G (x) + θ2 sin (πG (x))

(1.1.470)

x ∈ R, where α ≥ 0, β ≥ 0, γ ≥ 0, θ ≥ 0, α + γ > 0, α + β > 0 are parameters and G (x) is a baseline cd f with the corresponding pd f g (x).

1.1.234

Generalized Burr XII Power Series (GBXIIPS)

The cd f and pd f of GBXIIPS are given, respectively, by   λ  C θ 1 − (1 + xα )−µ F(x; α, θ, λ, µ) = , x ≥ 0, C (θ)

(1.1.471)

and d F(x; α, θ, λ, µ), x > 0, (1.1.472) dx n where α, λ, µ are all positive parameters and C (θ) = ∑∞ n=1 an θ is finite for an ≥ 0 and θ > 0. f (x; α, θ, λ, µ) =

Remark 1.176. Nasir et a. (2019) proposed the following distribution iα   h C λ − λ 1 − (1 + xc )−k F (x; α, λ, c, k) = 1 − , x ≥ 0. C (λ) Clearly, the characterizations of the cd f (1.1.471) are similar to those of F (x; α, λ, c, k) which is numbered (1.1.143) in this work.

G. G. Hamedani

118

1.1.235

Transmuted Extended Exponential (TEE)

The cd f and pd f of TEE are given, respectively, by

and

h i h i α α 2 F(x; α, β, λ) = (1 + λ) 1 − e1−(1+βx) − λ 1 − e1−(1+βx) , x ≥ 0,

d F(x; α, β, λ), x > 0, dx where α > 0, β > 0, |λ| ≤ 1 are parameters. h i α Remark 1.177. Taking G (x) = 1 − e1−(1+βx) , x ≥ 0, we have f (x; α, β, λ) =

F (x; α, β, λ) = (1 + λ) G (x) − λG (x)2 ,

(1.1.473)

(1.1.474)

x ≥ 0,

which is considered in Remark 1.27.

1.1.236

Exponentiated Generalized Power Series (EGPS)

The cd f and pd f of EGPS are given, respectively, by  h  c i C λ 1 − 1 − (1 − G (x))d F(x; c, d, λ) = 1 − , x ∈ R, C (λ)

(1.1.475)

and d F(x; c, d, λ), x ∈ R, (1.1.476) dx n where c > 0, d > 0 are parameters and and C (λ) = ∑∞ n=1 an λ is finite for an ≥ 0 and λ > 0. f (x; c, d, λ) =

Remark 1.178. Condino and Domma (2017) proposed the following distribution F (x; α, θ, ν) = 1 −

C (θ [1 − G1 (x)]) , x ∈ R, C (θ)

where α, θ, ν are all positive parameters and G1 (x) is a baseline cd f . Taking G1 (x) =  c 1 − (1 − G (x))d , x ∈ R, we arrive at the cd f (1.1.475). The cd f F (x; α, θ, ν) has been characterized in Hamedani (2017).

1.1.237

Marshall-Olkin Alpha Power Inverse Exponential (MOAPIE)

The cd f of MOAPIE is given by  −λx−1  αe   −1 −1  , (α−1)θ−(1−θ) αe−λx −1 F(x; α, θ, λ) =  e−λx−1 ,

α 6= 0 α=1

, x ≥ 0,

Introduction

119

and f (x; α, θ, λ) =

d F(x; α, θ, λ), x > 0, dx

where α, θ, λ are all positive parameters. Remark 1.179. We will characterize the following distribution called Marshall-Olkin Al−1 pha Power (MOAP) of which MOAPIE is a special case for G (x) = e−λx , x ≥ 0.

and

The cd f and pd f of MOAP are given, respectively, by  αG(x) −1  , α 6= 0 (α−1)θ−(1−θ)(αG(x) −1) F(x; α, θ, λ) = , x ∈ R, G(x), α=1 f (x; α, θ) =

 

(α−1)θ log(α)αG(x) g(x)

[(α−1)θ−(1−θ)(αG(x) −1)] g(x),

2

,

α 6= 0 α=1

, x ∈ R,

(1.1.477)

(1.1.478)

where α, θ are positive parameters and G (x) is a baseline cd f with the corresponding pd f g (x).

1.1.238

Transmuted Topp Leone Exponentiated Fréchet (TTLEFr)

The cd f and pd f of TTPEFr are given, respectively, by  n o2 α −θ(a/x)b F(x; α, θ, λ, a, b) = (1 + λ) 1 − 1 − e − n o2 2α −θ(a/x)b , λ 1− 1−e 

x ≥ 0, and

(1.1.479)

d F(x; α, θ, λ, a, b), x > 0, dx where α, θ, a, b all positive and λ ∈ [−1, 1] are parameters.  n o α b 2 −θ(a/x) Remark 1.180. Taking G (x) = 1 − 1 − e , x ≥ 0, we have f (x; α, θλ, a, b) =

F(x; α, θ, λ, a, b) = (1 + λ) G (x) − λG (x)2 ,

which is considered in Remark 1.27.

x ≥ 0,

(1.1.480)

G. G. Hamedani

120

1.1.239

Generalized Gudermannian (GG)

The cd f and pd f of GG are given, respectively, by F(x; β) = and

  2 arctan eβx , x ∈ R, π

f (x; β) = where β is a positive parameter.

1.1.240

2βeβx
, x ∈ R, π 1 + e2βx

(1.1.481)

(1.1.482)

Alpha Skew Generalized Gudermannian (ASGG)

The cd f and pd f of ASGG are given, respectively, by F(x; α) =

Z x

−∞

f (u; α)du, x ∈ R,

(1.1.483)

and f (x; α) =

h

i (1 − αx)2 + 1 eπx/2 (2 + α2 )(1 + eπx )

, x ∈ R,

(1.1.484)

where α is a positive parameter.

1.1.241

Truncated-Logistic Skew-Symmetric (TLSS)

The cd f and pd f of TLSS are given, respectively, by " # ! 1 − e−λG(x) 1 + e−λ F(x; λ) = , x ∈ R, 1 + e−λG(x) 1 − e−λ

(1.1.485)

and "

λg (x) e−λG(x) f (x; λ) = (1 + e−λG(x) )2

#

2(1 + e−λ ) 1 − e−λ

!

, x ∈ R,

(1.1.486)

where λ ∈ R is a parameter and G (x) is a baseline cd f with the corresponding pd f g (x).

1.1.242

Combined Exponential-Normal {Generalized Weibull} (CE-N{GW})

The cd f and pd f of CE-N{GW} , WLOG for µ = 0, σ = 1, are given, respectively, by  n o  F(x; λ) = 1 − exp − (1 − Φ (x))−λ /λ , x ∈ R, (1.1.487)

and

Introduction

f (x; λ) =

121

 n o  φ (x) exp − (1 − Φ (x))−λ /λ (1 − Φ (x))λ+1

, x ∈ R,

(1.1.488)

where λ > 0 is a parameter and Φ (x) is the cd f of the standard normal random variable with the corresponding pd f φ (x).

1.1.243

Type II Half Logistic Ibrahim (TIIHLI)

The cd f and pd f of TIIHLI are given, respectively, by F(x; λ) =

2 [G (x)]λ 1 + [G (x)]λ

, x ≥ 0,

(1.1.489)

and d F(x; λ), x > 0, (1.1.490) dx   2 1 where λ > 0 is a parameter and G (x) = 1 − θ+1 e−βx + θe−(βx) , x ≥ 0 , is a baseline cd f . f (x; λ) =

Remark 1.181. Hassan et al. (2017) proposed the following distribution F (x; λ, ζ) =

2G (x; ζ)λ 1 + G (x; ζ)λ

, x ∈ R,

which is the same as cd f (1.1.489) for x ∈ R. The cd f F (x; λ, ζ) has been characterized in Hamedani (2018a).

1.1.244

Modified Generalized Marshall-Olkin (MGMO)

The cd f and pd f of MGMO are given, respectively, by ( ) θG (x; η)λ 1 − G (x; η)λ F(x; θ, λ, η) = 1+ , 1 + (θ − 1) G (x; η)λ 1 + (θ − 1) G (x; η)λ

(1.1.491)

x ∈ R, and d F(x; θ, λ, η), x ∈ R, dx where θ > 0, λ > 0 are parameters and G (x; η) is a baseline cd f . f (x; θ, λ, η) =

Remark 1.182. Hamedani, (2017) characterized the following distribution   θK (x; φ) γK (x; φ) F (x) = 1+ , x ∈ R. 1 + (θ − 1) K (x; φ) 1 + (θ − 1) K (x; φ)

(1.1.492)

Taking γ = 1 and K (x; φ) = G (x; η)λ , we arrive at the cd f (1.1.491). So, the proposed cd f (1.1.491) is not new.

G. G. Hamedani

122

1.1.245

Transmuted Extended Lomax (TEL)

The cd f and pd f of TEL are given, respectively, by   α F (x; α, β, λ, γ) = (1 + λ) 1 − (1 + βx)γ − (1 − α)  2 α −λ 1− , (1 + βx)γ − (1 − α)

(1.1.493)

x ≥ 0, and d F(x; α, β, λ, γ), x > 0, dx where α, β, γ all positive and |λ| ≤ 1 are parameters. f (x; α, β, λ, γ) =

(1.1.494)

Remark 1.183. Bourguignon et al. (2016) introduced the following distribution F (x; λ, η) = (1 + α) G (x; η) − λ [G (x; η)]2 , x ∈ R. n o Taking G (x; η) = 1 − (1+βx)γα−(1−α) , x ≥ 0 in F (x; λ, η), we arrive at the cd f (1.1.493). The cd f F (x; λ, η) has been characterized in Hamedani and Maadooliat (2017).

1.1.246

Topp-Leone Odd Log-Logistic Exponential (TLOLLEx)

The cd f and pd f of TLOLLEx are given, respectively, by

x ≥ 0, and

 #2 b " a    −λx 1−e a F (x; a, b, λ) = 1 − 1 − −λax  ,   e + 1 − e−λx f (x; a, b, λ) =

d F(x; a, b, λ), x > 0, dx

(1.1.495)

(1.1.496)

where a, b, λ are all positive parameters. Remark 1.184. Brito et al. (2017) introduced the following distribution 2 )b G (x; λ)a F (x; a, b, λ) = 1 − 1 − , x ∈ R. G (x; λ)a + G (x; λ)a   Taking G (x; λ) = 1 − e−λx , x ≥ 0 in F (x; a, b, λ), we arrive at the cd f (1.1.495). The cd f F (x; a, b, λ) has been characterized in Hamedani (2018b). (



Introduction

1.1.247

123

Odd Birnbaum-Saunders (OBS)

The cd f and pd f of OBS are given, respectively, by F (x; α, λ, η) =

Φα (ν) , x ≥ 0, Φα (ν) + Φ (−ν)α

(1.1.497)

and n o τ(x/η) ακ (λ, η) x−3/2 (x + η) exp − 2λ2 f (x; α, λ, η) =  , 2 Φα (ν) + Φ (−ν)α [Φ (ν)Φ (−ν)]1−α

x > 0, where α, λ, η all positive are parameters, κ (λ, β) =

−2

λ e√ , τ (u) 2λ 2πβ

(1.1.498) = u + u−1 , ν =

λ−1 ρ (x/β) , ρ (u) = u1/2 − u−1/2 and Φ (·) is the standard normal cumulative distribution function. Remark 1.185. The cd f of OBSG (see cd f (1.1.247), Hamedani (2019)) is given by F (x; α, λ, β, p) =

Φα (ν) , x ≥ 0, Φα (ν) + (1 − p) [1 − Φ (ν)]α

where α, λ, β all positive, p ∈ (0, 1) are parameters, κ (λ, β) =

−2

λ e√ , τ (u) = u + u−1 2λ 2πβ

,ν=

λ−1 ρ (x/β) , ρ (u) = u1/2 − u−1/2 and Φ (·) is the standard normal cumulative distribution function. Characterizations of the cd f OBS are similar to those of OBSG which are given in Hamedani (2019).

1.1.248

Gamma Dual Weibull (Γ2 -W)

The cd f and pd f of Γ2 -W are given, respectively, by    β γ a, − log 1 − e−αx F (x; α, β, a) = 1 − , Γ (a)

(1.1.499)

x ≥ 0, and f (x; α, β, a) =

β  ia−1 αβxβ−1 e−αx h β − log 1 − e−αx , Γ (a)

(1.1.500)

x > 0, where α, β, a are all positive parameters.

1.1.249

Quasi XGamma-Poisson (QXGP)

The cd f and pd f of QXGP are given, respectively, by

F (x; α, θ, λ) = and

eλ − e

λe−θx 1+α+θx+ 12 θ2 x2

(

α+1

eλ − 1

) ,

(1.1.501)

G. G. Hamedani

124

d F (x; α, θ, λ), x > 0, dx where α, θ, λ are all positive parameters. f (x; α, θ, λ) =

(1.1.502)

Remark 1.186. Nadarajah et al. (2009) proposed the following distribution F (x) =

1 − e−λG(x) eλ − eλG(x) = , x ∈ R, 1 − e−λ eλ − 1

e−θx (1+α+θx+ 21 θ2 x2 ) where G (x) is a baseline cd f . Taking G (x) = 1 − , x ≥ 0, in F (x), we α+1 arrive at cd f (1.1.501). The characterizations of the cd f (1.1.501) are similar to those of F (x) which have appeared in Hamedani (2016).

1.1.250

Modified Extended Generalized Exponential (MEGE)

The cd f and pd f of MEGE are given, respectively, by F (x; α, β, η, λ) = and

β − ηe−λx


α/η

− (β − η)α/η

βα/η − (β − η)α/η

, x ≥ 0,

d F (x; α, β, η, λ), x > 0, dx where α, β, η, λ are all positive parameters. f (x; α, β, η, λ) =

(1.1.503)

(1.1.504)

Remarks 1.187. (a) Clearly, the condition β > η is missing. (b) A simple change of variables Y = β − ηe−λX , x ≥ 0, shows that Y has a power function distribution on [0, β]. The power function distribution has been around for a long time and has been characterized in many directions by Hamedani.

1.1.251

Exponentiated Nadarajah Haghighi Poisson (ENHP)

The cd f and pd f of ENHP are given, respectively, by 1−(1+ωx)α β

and

1 − e−λ(1−e F (x; α, β, ω, λ) = 1 − e−λ

)

, x ≥ 0,

d F (x; α, β, ω, λ), x > 0, dx where α, β, ω, λ are all positive parameters. f (x; α, β, ω, λ) =

(1.1.505)

(1.1.506)

Introduction

125

Remarks 1.188. (a) Nadarajah et al. (2009) proposed the following distribution 1 − e−λG(x) , x ∈ R, 1 − e−λ   α β where G (x) is a baseline cd f . Taking G (x) = 1 − e1−(1+ωx) , x ≥ 0, in F (x), we arrive at cd f (1.1.505). The cd f F (x) has been characterized in Hamedani (2016). (b) The cd f (1.1.189) is given by n  α o λ 1 − exp −θ 1 − e−β(1−e )x , x ≥ 0, F (x; α, β, θ, λ) = 1 − e−θ F (x) =

which, in a way, more general than the cd f (1.1.505). (c) The cd f (1.1.111) by   2α υ −λ 1−e−x

1−e 1 − e−λ which, in a way, more general than the cd f (1.1.505). F (x; λ, υ, α) =

1.1.252

, x ≥ 0,

Extended New Generalized Exponential (ENGE)

The cd f and pd f of ENGE (as copied by the authors) are given, respectively, by
e−λαx − 1
F (x; α, β, λ) = α β − (β − 1)α ( )   α α α ∑ (−1) k βα−k , x ≥ 0, k=0

(1.1.507)

and d F (x; α, β, λ), x > 0, dx where α > 1, β > 1, λ > 0 are all positive parameters. f (x; α, β, λ) =

Remark 1.189. The proposed cd f (1.1.507) or )
( α   e−λαx − 1 α
∑ (−1)k F (x; α, β, λ) = α βα−k , (∗) k β − (β − 1)α k=0

neither is a cd f and here is why: Taking α = β = 2, then (1.1.507) will be )     2 2 2−k −λ2x (−1) 2 e − 1 ∑ k k=0     1 = {4 + 4 + 1} e−λ2x − 1 = 3 e−λ2x − 1 , 3

1 F(x) = 3

(

2

(1.1.508)

G. G. Hamedani

126

which is not a cd f . With the same α and β, (∗) will be )     2 k 2−k −λ2x (−1) e 2 − 1 ∑ k k=0   1  1 = {4 − 4 + 1} e−λ2x − 1 = e−λ2x − 1 , 3 3

1 F (x) = 3

(

2

which is not a cd f . It should be mentioned that this Remark was sent to one of the authors.

1.1.253

Transmuted Singh-Maddala (TSM)

The cd f and pd f of TSM are given, respectively, by   α −δ #  α −δ #" x x 1+λ 1+ = F (x; α, β, δ, λ) = 1 − 1 + β β   α −2δ   α −δ x x 1 − (1 − λ) 1 + −λ 1+ , x ≥ 0, β β "



(1.1.509)

and d F (x; α, β, δ, λ), x > 0, dx where α > 0, β > 0, λ ∈ [−1, 1] are parameters. f (x; α, β, δ, λ) =

(1.1.510)

Remark 1.190. Al-Khazaleh (2016) proposed the following cd f h  x c i−k  h  x c i−k  F (x; c, k, s, λ) = 1 − 1 + 1−λ+λ 1+ , x ≥ 0, s s

which is exactly the cd f (1.1.509). It is quite possible that this distribution was proposed independently by Al-Khazaleh (2016) and Shahzad et al. (2017). The cd f F (x; c, k, s, λ) has been characterized in Hamedani and Safavimanesh (2017).

1.1.254

Transmuted Half Logistic (THL)

The cd f and pd f of THL are given, respectively, by F (x; λ) =

(ex − 1) (1 + 2λ + ex ) (1 + ex )2

, x ≥ 0,

(1.1.511)

and f (x; λ) = where λ ∈ [−1, 1] is a parameter.

2ex {(1 − λ) ex + 3λ + 1} (1 + ex )3

, x > 0,

(1.1.512)

Introduction

127

Remark 1.191. The cd f (1.1.511) can be expressed as

x

F (x; λ) = (1 + λ) G (x) − λG2 (x) ,

−1 , x ≥ 0, which has been characterized in our previous work. That being where G (x) = eex +1 said, we will characterized (1.1.511) in Subsections 2.1 and 2.3 below.

1.1.255

Kumarswamy Exponentiated U-Quadratic (KwEUQ)

The cd f and pd f of KwEUQ are given, respectively, by   α θλ  θλ δ F (x; α, β, θ, δ, λ, a, b) = 1 − 1 − (x − β)3 + (β − a)3 , 3

(1.1.513)

x ∈ [a, b], and

f (x; α, β, θ, δ, λ, a, b) = x ∈ (a, b), where a ∈ R, b ∈ (a, ∞) , α = eters.

d F (x; α, β, θ, δ, λ, a, b), dx

(12) (b−a)3

> 0, β =

b+a 2 ,θ

(1.1.514)

> 0, δ > 0, λ > 0 are param-

Remark 1.192. Characterizations of the cd f (1.1.513) are similar to those of the cd f (1.1.39).

1.1.256

New Generalized Transmuted Inverse Exponential (NGT-IE)

The cd f and pd f of NGT-IE are given, respectively, by h  α i F (x; α, β, λ) = (1 + λ) 1 − 1 − e−β/x h  α i2 − λ 1 − 1 − e−β/x , x ≥ 0,

(1.1.515)

and

d F (x; α, β, λ), x > 0, dx where α > 0, β > 0 and λ ∈ [−1, 1] are parameters. f (x; α, β, λ) =

Remarks 1.193. (a) Khan et al. (2017) proposed the following distribution   " (  β )#φ   α 1 F (x; α, β, γ, φ, λ) = 1 − 1 − exp − − γ ×   x x   " (  β )#φ   α 1 1 + λ 1 − exp − − γ , x ≥ 0.   x x

(1.1.516)

G. G. Hamedani

128

Clearly F (x; α, β, γ, φ, λ) seems more general, in some way, than the cd f (1.1.515). The distribution F (x; α,
αβ, γ, φ, λ) has been characterized in Hamedani (2018a). (b) Taking −β/x G (x) = 1 − 1 − e , x ≥ 0, in the cd f (1.1.515), we arrive at F (x; α, β, γ, φ, λ) = (1 + λ) G (x) − λG2 (x) ,

which has been characterized in Hamedani (2017).

1.1.257

Transmuted Lomax Exponential (TLE)

The cd f and pd f of TLE are given, respectively, by    2 F (x; α, β, θ, λ) = (1 + λ) 1 − βα (β + θx)−α − λ 1 − βα (β + θx)−α ,

(1.1.517)

x ≥ 0, and

d F (x; α, β, θ, λ), x > 0, dx where α > 0, β > 0, θ > 0 and λ ∈ [−1, 1] are parameters. f (x; α, β, θ, λ) =

(1.1.518)

Remark 1.194. Taking G (x) = 1 − βα (β + θx)−α , x ≥ 0, in the cd f (1.1.517), we arrive at F (x; α, β, θ, λ) = (1 + λ) G (x) − λG2 (x) , which has been characterized in Hamedani (2017).

1.1.258

Extended Pranav (EP)

The cd f and pd f of EP are given, respectively, by ( " )α
# θx θ2 x2 + 3θx + 6 F (x; α, θ) = 1 − 1 + e−θx , x ≥ 0, θ4 + 6

(1.1.519)

and
αθ4 θ + x3 e−θx f (x; α, θ) = × θ4 + 6 ( " )α−1
# θx θ2 x2 + 3θx + 6 1− 1+ e−θx , θ4 + 6 x > 0, where α > 0 and θ > 0 are parameters.

(1.1.520)

Introduction

1.1.259

129

Generalized Odd Lindley-G (GOLi-G)

The cd f and pd f of GOLi-G are given, respectively, by (

λG (x; ϕ)α   F (x; α, λ, ϕ) = 1 − 1 + (1 + λ) 1 − G (x; ϕ)α   −λG (x; ϕ)α exp , x ∈ R, 1 − G (x; ϕ)α

)

× (1.1.521)

and   −λG (x; ϕ)α αλ2 g (x; ϕ) G (x; ϕ)α−1 f (x; α, λ, ϕ) = , x ∈ R,  3 exp 1 − G (x; ϕ)α (1 + λ) 1 − G (x; ϕ)α

(1.1.522)

where α > 0 and λ > 0 are parameters and G (x; ϕ) is a baseline cd f with the corresponding pd f g (x; ϕ), which may depend on the parameter vector ϕ.

1.1.260

Leaned Normal (LN)

The cd f and pd f of LN are given, respectively, by F (x; α, β) = and

[Φ (x)]α
, x ∈ R, 1 − (1 − β) 1 − [Φ (x)]α

αβφ (x) [Φ (x)]α−1 f (x; α, λ, ϕ) = 
2 , x ∈ R, 1 − (1 − β) 1 − [Φ (x)]α

(1.1.523)

(1.1.524)

where α > 0 and β > 0 are parameters and Φ (x) is the standard normal cd f with the corresponding pd f φ (x). Remark 1.195. Hamedani and Risti´c (Marshall-Olkin Extended Distribution, 2016) proposed their distribution based on the cd f given below 

γK (x) F (x; γ, δ) = 1 − 1 − γK (x)

δ

, x ∈ R,

where K (x) is a baseline cd f . Taking δ = 1 in F (x; γ, δ), we arrive at F (x; γ, 1) =

K (x) , x ∈ R. 1 − γ(1 − K (x))

Letting K (x) = [Φ (x)]α , x ∈ R in F (x; γ, 1), we have the cd f (1.1.523). The cd f F (x; γ, δ) has been characterized in Hamedani and Risti´c (2016).

G. G. Hamedani

130

1.1.261

XGamma (XG)

The cd f and pd f of XG are given, respectively, by   2 2 1 + θ + θx + θ 2x F (x; θ) = 1 − e−θx , x ≥ 0, 1+θ and   θ2 θ 2 −θx 1 + x e , x ≥ 0, f (x; θ) = 1+θ 2

(1.1.525)

(1.1.526)

where θ > 0 is a parameter.

Remarks 1.196. (a) Ghitany et al. (2013) introduced the following distribution   1 + β + βxα exp [−βxα ], x ≥ 0, F (x; α, β) = 1 − 1+β

which is quite similar to (1.1.525). The cd f F (x; α, β) has been characterized in Hamedani and Maadooliat (2017). (b) Yousof et al. (2019) proposed the following cd f (which is cd f (1.1.243))   2 2b b 1 + θ + θxb + θ 2x e−θx F (x; θ, b) = 1 − , x ≥ 0, 1+θ which is a generalization of the cd f (1.1.525).

1.1.262

Quasi XGamma-Geometric (QXGGc)

The cd f and pd f of QXGGc are given, respectively, by " #   2 F (x; α, θ, p) = "

1−

1−

1+α+θx+ θ2 x2 e−θx 1+α

#,   2 p 1+α+θx+ θ2 x2 e−θx

x ≥ 0,

(1.1.527)

1+α

and d F (x; α, θ, p), x ≥ 0, dx where α > 0, θ > 0 and p ∈ (0, 1) are parameters. f (x; α, θ, p) =

(1.1.528)

Remark 1.197. Hamedani and Risti´c (Marshall-Olkin Extended Distribution, 2016) proposed their distribution based on the cd f given below 

γK (x) F (x; γ, δ) = 1 − 1 − γK (x)

δ

, x ∈ R,

where K (x) is a baseline cd f . Taking δ = 1 in F (x; γ, δ), we arrive at

Introduction

F (x; γ, 1) = 

131

K (x) , x ∈ R. 1 − γ(1 − K (x))

 2 1+α+θx+ θ2 x2 e−θx

, x ≥ 0 in F (x; γ, 1), we have the cd f (1.1.527). Letting K (x) = 1 − 1+α The cd f F (x; γ, δ) has been characterized in Hamedani and Risti´c (2016).

1.1.263

Marshall-Olkin Modified Burr III (MOMBIII)

The cd f and pd f of MOMBIII are given, respectively, by F (x; α, β, γ, λ) = and

1 + γx−β


−α/γ

λ + (1 − λ) 1 + γx−β

αβλx−β 1 + γx−β


−α/γ , x ≥ 0,


−α/γ−1

f (x; α, β, γ, λ) = h
−α/γi2 , x ≥ 0, −β λ + (1 − λ) 1 + γx

(1.1.529)

(1.1.530)

where α, β, γ, λ are all positive parameters.

Remark 1.198. Alizadeh et al. (2017) proposed the following distribution F (x) =

G (x; γ)α , x ∈ R, G (x; γ)α + βG (x; γ)α

where G (x; γ) is a baseline cd f . Taking α = 1 in F (x), we arrive at G (x; γ) G (x; γ) , x ∈ R. = G (x; γ) + βG (x; γ) β + (1 − β) G (x; γ)
−α/γ Now, Letting G (x; γ) = 1 + γx−β , x ≥ 0, we have the cd f (1.1.529). Alizadeh et al.’s distribution has been characterized in Hamedani (2019). F (x) =

1.1.264

Type II Exponentiated Half Logistic Generated (TIIEHL-G)

The cd f and pd f of TIIEHL-G are given, respectively, by " #α 1 − [G (x; η)]λ F (x; α, λ) = 1 − , x ∈ R, 1 + [G (x; η)]λ

(1.1.531)

and  α−1 2αλg (x; η) [G (x; η)]λ−1 1 − [G (x; η)]λ f (x; α, λ) = ,  α+1 1 + [G (x; η)]λ

(1.1.532)

x ∈ R, where α, λ are positive parameters and G (x; η) is a baseline cd f with corresponding pd f g (x; η), which may depend on the vector parameter η.

G. G. Hamedani

132

Remark 1.199. Hamedani and Risti´c (Marshall-Olkin Extended Distribution, 2016) proposed their distribution based on the cd f given below 

αK (x) F (x; α, δ) = 1 − 1 − αK (x)

where K (x) is a baseline cd f . Taking α =

1 2

δ

, x ∈ R,

in F (x; α, δ), we arrive at

    1 1 − K (x) δ F x; , δ = 1 − , x ∈ R. 2 1 + K (x)
Letting K (x) = [G (x; η)]λ , x ∈ R in F x; 12 , δ , we have the cd f (1.1.531). The cd f F (x; α, δ) has been characterized in Hamedani and Risti´c (2016).

1.1.265

Weighted XGamma (WXG)

The cd f and pd f of WXG are given, respectively, by 2θ F (x; θ, r) = × r! [2θ + (1 + r) (2 + r)]   1 γ (r + 1, θx) + γ (r + 3, θx) , x ≥ 0, 2θ

(1.1.533)

and   2θr+2e−θx θ r+2 r f (x; θ, r) = x + x , r! [2θ + (1 + r) (2 + r)] 2

(1.1.534)

x > 0, where θ > 0 is a parameter and r = 1, 2, ....

1.1.266

Length Biased XGamma (LBXG)

The cd f d an pd f of LBXG are given, respectively, by   (θ + 3) (1 + θx) + 32 θ2 x2 + 12 θ3 x3 −θx F (x; θ) = 1 − e , x ≥ 0, θ+3 and   θ3 θ 3 −θx f (x; θ) = x + x e , x > 0, (θ + 3) 2 where θ > 0 is a parameter.

(1.1.535)

(1.1.536)

Introduction

1.1.267

133

Wrapped XGamma (WRXG)

The cd f and pd f of WRXG are given, respectively, by F (x; λ) =

Z x 0

f (u; λ)du, x ∈ [0, 2π],

(1.1.537)

and λ2 e−λx
× (λ + 1) 1 − e−2πλ    2 1 + λx2 + 
2πλ (π − x) e−2π + (x + π)

f (x; λ) =



,

e−2πλ (1−e−2πλ)

x ∈ (0, 2π), where λ > 0 is a parameter.

1.1.268

(1.1.538)

Linearly Decreasing Stress Weibull (LDSWeibull)

The cd f and pd f of LDSWeibull are given, respectively, by   (x − θ)γ , x ≥ θ, F (x; θ, γ, τ) = 1 − exp − τx

(1.1.539)

and

f (x; θ, γ, τ) =

  (x − θ)γ−1 [(γ − 1) x + θ] (x − θ)γ exp − , τx2 τx

x > θ,

(1.1.540)

where θ ≥ 0, γ > 1, τ > 0 are parameters.

1.1.269

L-Logistic (L-Logistic)

The cd f and pd f of L-Logistic are given, respectively, by F (x; m, b) = and

xb (1 − m)b

xb (1 − m)b + mb (1 − x)b

, 0 ≤ x ≤ 1,

b (1 − m)b mb xb−1 (1 − x)b−1 f (x; m, b) = h i2 , 0 < x < 1, xb (1 − m)b + mb (1 − x)b

where b > 0, 0 < m < 1 are parameters.

(1.1.541)

(1.1.542)

G. G. Hamedani

134

1.1.270

Normal Generalized Hyperbolic Secant (NGHS)

The cd f and pd f of NGHS, WLOG for α = 0 , β = 1, are given, respectively, by    π  1 γ + sinh x , x ∈ R, (1.1.543) F (x; γ, ν) = Φ ν 2

and

π  1   π  π f (x; γ, ν) = cosh x φ γ + sinh x , x ∈ R, 2ν 2 ν 2

(1.1.544)

where γ > 0, ν > 0 are parameters and Φ (x) is the cd f of the standard normal with the corresponding pd f φ (x).

1.1.271

Generalized Inverse Pareto-G (GIP-G)

The cd f and pd f of GIP-G, after simplification, are given, respectively, by  a G (x; ν)c F (x; a, b, c, ν) = , x ∈ R, b + (1 − b) G (x; ν)c

(1.1.545)

and

d F (x; a, b, c, ν), x ∈ R, dx where a, b, c, ν are all positive parameters and G (x; ν) is a baseline cd f . f (x; a, b, c, ν) =

Remark 1.200. Taking γ =

1.1.272

1 b

(1.1.546)

in the cd f (1.1.241), we arrive at the cd f (1.1.545).

Kumarsawamy Marshall-Olkin Modified Weibull (KMOMW)

The cd f and pd f of KMOMW are given, respectively, by ( !) β 1 − e−(θx+γx ) F (x; β, γ, θ, a, b, p) = 1 − 1 − , x ≥ 0, β 1 − pe−(θx+γx )

(1.1.547)

and

d F (x; β, γ, θ, a, b, p), x > 0, dx where β > 0, γ ≥ 0, θ ≥ 0, a > 0, b > 0, p > 0 are parameters. f (x; β, γ, θ, a, b, p) =

(1.1.548)

β Remark 1.201. Taking G (x) = 1 − e−(θx+γx ), x ≥ 0 and λ = 1, the cd f (1.1.439) will reduce to the cd f (1.1.547).

Introduction

1.1.273

135

Wrapped Lindley-Exponential (WRLE)

The cd f and pd f of WRLE are given, respectively, by   αβx −αβx 1 −αβx e 1 − e − (1 + α) 1 − e−2παβ h i −2παβ 2παβe −αβx 1 − e , +
2 (1 + α) 1 − e−2παβ

F (x; α, β) =

and

f (x; α, β) =

d F (x; α, β), x > 0, dx

(1.1.549)

(1.1.550)

where α > 0, β > 0 are parameters. Remark 1.202. Characterizations of the cd f (1.1.549) are similar to those of the cd f (1.1.537).

1.1.274

Odoma

The cd f and pd f of Odoma are given, respectively, by   θ2 x2 (θ2 x2 +4θx+12) + 1+ θ2 +θ3 +24  e−θx , x ≥ 0, F (x; θ) = 1 −  θx(θ4 x+2θ3 +48)

(1.1.551)

2(θ2 +θ3 +24)

and


θ5 2x4 + θx2 + 2θ e−θx f (x; θ) = , x > 0, 2(θ2 + θ3 + 24)

(1.1.552)

where θ > 0 is a parameter.

1.1.275

Wibull Inverse Lomax (WIL)

The cd f and pd f of WIL are given, respectively, by  " #−b  λ    β F (x; β, λ, a, b) = 1 − exp −a 1 + −1 ,   x

(1.1.553)

x ≥ 0, and

d F (x; β, λ, a, b), x > 0, dx where β, λ, a, b are all positive parameters. f (x; β, λ, a, b) =

(1.1.554)

Remark 1.203. The cd f (1.1.553) is a special case of the cd f (1.1.435) proposed by Falgore et al. (2019). It is possible that these two works were done independently.

G. G. Hamedani

136

1.1.276

Poisson Burr X Pareto Type II (PBXPTII)

The cd f and pd f of PBXPTII are given, respectively, by

F (x; β, λ, θ) =

"

#  h i2 θ β 1 − exp −λ 1 − exp − (1 + x) − 1 

1 − e−λ

, x ≥ 0,

(1.1.555)

and f (x; β, λ, θ) =

d F (x; β, λ, θ), x > 0, dx

(1.1.556)

where β, λ, θ are all positive parameters.  h i2  β Remark 1.204. Taking G (x) = 1 − exp − (1 + x) − 1 , x ≥ 0, the cd f (1.1.555) can

be expressed as

θ

1 − e−λG(x) , F (x; β, λ, θ) = 1 − e−λ which has been mentioned in Remark 1.40.

1.1.277

Burr XII Fréchet (BrXIIFr)

The cd f and pd f of BrXIIFr are given, respectively, by F (x; α, β, a, b) = 1 − 1 +

(

and

b

e−(a/x)

b

1 − e−(a/x)

)α !−β

, x ≥ 0,

d F (x; α, β, a, b), x > 0, dx where α, β, a, b are all positive parameters. f (x; α, β, a, b) =

b

(1.1.557)

(1.1.558)

Remark 1.205. Taking G (x) = e−(a/x) , x ≥ 0, the cd f (1.1.557) can be expressed as G (x) F (x; α, β, a, b) = 1 − 1 + G (x) which is the cd f (1.1.57).





α −β

,

Introduction

1.1.278

137

Burr X Fréchet (BrXFr)

The cd f and pd f of BrXFr are given, respectively, by  ( )2 θ b −(a/x) e  , x ≥ 0, F (x; θ, a, b) = 1 − exp − b 1 − e−(a/x) 

and

d F (x; α, β, a, b), x > 0, dx where θ, a, b are all positive parameters. f (x; θ, a, b) =

(1.1.559)

(1.1.560)

b

Remark 1.206. Taking G (x) = e−(a/x) , x ≥ 0, the cd f (1.1.559) can be expressed as F (x; θ, a, b) =

"   #!θ G (x) 2 , 1 − exp − G (x)

which is the cd f (1.1.7). Please see Remark 1.2 as well.

1.1.279

Odd Generalized Exponential Type-I Generalized Half Logistic (OGET-IGHL)

The cd f and pd f of OGET-IGHL are given, respectively, by  β !   ex/σ + 1 θ  F (x; α, β, σ, θ) = 1 − exp −α − 1 , x ≥ 0,   2 

and



(1.1.561)

d F (x; α, β, a, b), x > 0, (1.1.562) dx where α, β, σ, θ are all positive parameters.  x/σ −θ Remark 1.207. Taking K (x) = 1 − e 2+1 , x ≥ 0, in the cd f given in Remark 1.2, we arrive at the cd f (1.1.561). f (x; α, β, σ, θ) =

1.1.280

Burr X Nadarajah Haghighi (BXNH)

The cd f and pd f of BXNH are given, respectively, by   β ! 2  1 − exp 1 − (λx + 1)α  θ       F (x; α, θ, λ) = 1 − exp −  ,   exp 1 − (λx + 1)α 

(1.1.563)

G. G. Hamedani

138 x ≥ 0, and

d F (x; α, θ, λ), x > 0, (1.1.564) dx where α, θ, λ are all positive parameters.   Remark 1.208. Taking K (x) = 1 − exp 1 − (λx + 1)α , x ≥ 0, in the cd f given in Remark 1.208, we arrive at the cd f (1.1.563). f (x; α, θ, λ) =

1.1.281

Topp-Leone Compound Rayleigh (TLCR)

The cd f and pd f of TLCR are given, respectively, by h
−2αiθ F (x; α, β, θ) = 1 − β2α β + x2 ,

and

x ≥ 0,

d F (x; α, β, θ), x > 0, dx where α, β, θ are all positive parameters. f (x; α, β, θ) =

(1.1.565)

(1.1.566)

Remark 1.209. The cd f (1.1.565) can be written as "

x2 F (x; α, β, θ) = 1 − 1 + β 

−2α#θ

,

which is similar to the cd f given in Remark 1.108.

1.1.282

Dual Exponentiated Weibull (DEW)

The cd f and pd f of DEW are given, respectively, by   h iα λ θ β   −x      1−e    , F (x; α, β, λ, θ) = 1 − exp −   β α    1 − 1 − e−x   

and

x ≥ 0,

(1.1.567)

d F (x; α, β, λ, θ), x > 0, (1.1.568) dx where α, β, λ, θ are all positive parameters. h i β α Remark 1.210. Taking K (x) = 1 − e−x , x ≥ 0, in the cd f given in Remark 1.2, we arrive at the cd f (1.1.567). f (x; α, β, λ, θ) =

Introduction

1.1.283

139

unit-Improved Second-Degree Lindley (unit-ISDL)

The cd f and pd f of unit-ISDL are given, respectively, by  
2(λ2 +λ)x x 2 + 1−x λ2 1−x λx  e− 1−x , 0 ≤ x ≤ 1, F (x; λ) = 1 − 1 + λ2 + 2λ + 2

(1.1.569)

and

λ3 (1 − x)−4 − λx f (x; λ) = 2 e 1−x , 0 < x < 1, λ + 2λ + 2

(1.1.570)

where λ > 0 is a parameter.

1.1.284

Kumaraswamy Odd Lindley-G (KOL-G)

The cd f and pd f of KOL-G are given, respectively, by

F (x; λ, η, a, b) =

 a !b G (x; η) λ + G (x; η) exp −λ , 1− 1− (1 + λ)G (x; η) G (x; η) 

(1.1.571)

x ≥ 0, and d F (x; λ, η, a, b), x > 0, dx where λ, η, a, b are all positive parameters. f (x; λ, η, a, b) =

(1.1.572)

Remark 1.211. Characterizations of the cd f (1.1.571) are similar to those of the cd f (1.1.351).

1.1.285

Transmuted Type I Generalized Logistic (TTIGL)

The cd f and pd f of TTIGL are given, respectively, by b

F (x; λ, b) =

(1 + λ) (1 + e−x ) − λ (1 + e−x )2b

,

x ∈ R,

(1.1.573)

and b

f (x; λ, b) =

be−x {(1 + λ) (1 + e−x ) − 2λ} (1 + e−x )2b+1

where λ ∈ [−1, 1] and b > 0 are parameters.

,

x ∈ R,

(1.1.574)

G. G. Hamedani

140

1.1.286

Truncated Discrete Linnik Weibull (TDLW)

The cd f and pd f of TDLW are given, respectively, by   i β α ν p + (1 − p) 1 − e−(λx) − pν F (x; α, ν, p, λ, β) = h   iν , β α (1 − pν ) p + (1 − p) 1 − e−(λx) h

and

x ≥ 0,

  β β α−1 ανβλβ pν (1 − p) xβ−1 e−(λx) 1 − e−(λx) f (x; α, ν, p, λ, β) = h   iν+1 , x > 0, β α (1 − pν ) p + (1 − p) 1 − e−(λx)

(1.1.575)

(1.1.576)

where α, ν, λ, β all positive and p (0, 1) are parameters.

1.1.287

Raised Cosine (RC)

The cd f and pd f of RC (WLOG for µ = 0 and σ = 1) are given, respectively, by   1 1 1 + x + sin (πx) , − 1 ≤ x ≤ 1, (1.1.577) F (x) = 2 π

and

f (x) =

1 [1 + cos (πx)] , 2

− 1 < x < 1.

(1.1.578)

Remark 1.212. Certain characterizations of the cd f (1.1.577) are given by Ahsanullah and Shakil (2018). The characterizations of the cd f (1.1.577) presented here are completely different from those of Ahsanullah and Shakil.

1.1.288

Transmuted Exponentiated Weibull (TEW)

The cd f and pd f of TEW are given, respectively, by

and

n o n   o β α β α F (x; α, β, η, λ) = 1 − e−ηx 1 + λ − λ 1 − e−ηx , x ≥ 0,

d F (x; α, β, η, λ), dx where α > 0, β > 0, η > 0, λ ∈ [−1, 1] are parameters. f (x; α, β, η, λ) =

x > 0,

Remark 1.213. The cd f (1.1.579) is a special case of the cd f (1.1.11).

(1.1.579)

(1.1.580)

Introduction

1.1.289

141

Generalized Marshall-Olkin Extended Burr-III (GMOBIII)

The cd f and pd f of GMOBIII are given, respectively, by

x ≥ 0, and

h 
−λ i θ  α 1 − 1 + x−β  h i F (x; α, β, θ, λ) = 1 − ,  1 − α 1 − 1 + x−β
−λ 

(1.1.581)

d F (x; α, β, θ, λ), x > 0, (1.1.582) dx where α, β, θ, λ are all positive parameters.
−λ Remark 1.214. Taking G (x) = 1 + x−β , x ≥ 0, the cd f (1.1.581) can be written as f (x; α, β, θ, λ) =

F (x; α, β, θ, λ) = 1 − which has been discussed in Remark 1.38.

1.1.290



αG (x) 1 − αG (x)

θ

,

Modified Beta Linear Exponential (MBLE)

The cd f and pd f of MBLE are given, respectively, by

F (x; β, λ, a, b, c) =

   −1  β 2 x +λx +c−1 ; a, b B 1 + −ce 2 B (a, b)

,

x ≥ 0,

(1.1.583)

and f (x; β, λ, a, b, c) =

d F (x; β, λ, a, b, c), dx

x > 0,

(1.1.584)

where β, λ, a, b, c are all positive parameters and B (x; a, b) = 0x t a−1 (1 − t)b−1 dt.  −1 β 2 Remark 1.215. Taking G (x) = 1 + −ce 2 x +λx + c − 1 , x ≥ 0, the cd f (1.1.583) can be written as R

F (x; β, λ, a, b, c) =

B (G (x) ; a, b) , B (a, b)

which is cd f (1.1.43) discussed in Remark 1.15.

G. G. Hamedani

142

1.1.291

Zero-Truncated Poisson-Power Function (ZTPPF)

The cd f and pd f of ZTPPF are given, respectively, by n
β o 1 − exp −λ αx , F (x; α, β, λ) = 1 − e−λ and f (x; α, β, λ) =

d F (x; α, β, λ), dx

x ≥ 0,

x > 0,

(1.1.585)

(1.1.586)

where α, β, λ are positive parameters. Remark 1.216. The cd f (1.1.585) is a special case of the cd f (1.1.111).

1.1.292

Cubic Transmuted Pareto (CTP)

The cd f and pd f of CTP are given, respectively, by  θ # "  θ  2θ # k k k F (x; θ, k, λ1 , λ2 ) = 1 − 1 + (λ1 + λ2 ) − λ2 , x x x "

x ≥ k, (1.1.587)

and d F (x; θ, k, λ1 , λ2 ) , x > k, (1.1.588) dx where θ > 0, k > 0, λ1 ∈ [−1, 1], λ2 ∈ [−1, 1] and −2 ≤ λ1 + λ2 ≤ 1 are parameters.
θ Remark 1.217. Taking G (x) = 1 − kx , x ≥ k , the cd f (1.1.587) can be expressed as h i F (x; θ, k, λ1, λ2 ) = G (x) 1 + λ1 + (λ2 − λ1 )G (x) − λ2 G (x)2 , f (x; θ, k, λ1, λ2 ) =

which has been characterized in Hamedani (2019).

1.1.293

Inverse XGamma (IXG)

The cd f and pd f of IXG are given, respectively, by   θ θ2 F (x; θ) = 1 + + e−θ/x , x ≥ 0, (1 + θ) x 2 (1 + θ) x2

(1.1.589)

and

θ2 f (x; θ) = (1 + θ) x2 where θ > 0 is a parameter.



 θ 1 + 2 e−θ/x , 2x

x > 0,

(1.1.590)

Introduction

1.1.294

143

Alpha-Power Transformed Lindley (αPTL)

The cd f and pd f of αPTL are given, respectively, by ( G(x) α −1 α−1 , if α > 0, α 6= 0 , F (x; α) = G(x), if α = 1

x ≥ 0,

(1.1.591)

and d F (x; α) , x > 0, dx where α > 0 is a parameter and G (x) is the Lindley distribution. f (x; α) =

(1.1.592)

Remark 1.218. The cd f (1.1.591), for a given baseline cd f has been characterized in Hamedani (2019). Similar characterizations can be stated for the αPTL distribution.

1.1.295

Log-Odd Logistic-Weibull (LOLW)

The cd f and pd f of LOLW are given, respectively, by

and

n h
γ ioα 1 − exp − λx F (x; α, γ, λ) = n h h
γioα n 1 − exp − λx + exp − f (x; α, γ, λ) =


x γ λ

ioα , x ≥ 0,

d F (x; α, γ, λ), x > 0, dx

(1.1.593)

(1.1.594)

where α, γ, λ are all positive parameters. Remark 1.219. The cd f (1.1.593) is the same as the cd f (1.1.281).

1.1.296

Skewed Generalized Logistic (SGL)

The cd f and pd f of SGL are given, respectively, by 1 F (x; k, λ, φ) = φB (k, k) and f (x; k, λ, φ) =

Z x

−∞

f (u)du, x ∈ R,

1 ekx/(1+sgn(x)λ)φ
, x ∈ R, φB (k, k) 1 + ekx/(1+sgn(x)λ)φ 2k

where k > 0, φ > 0 and λ ∈ [−1, 1] are parameters.

(1.1.595)

(1.1.596)

G. G. Hamedani

144

1.1.297

Weighted T-X (WTX)

The cd f and pd f of WTX are given, respectively, by F (x) = 1 − G (x) e−G(x) , x ∈ R,

(1.1.597)

f (x) = g (x) (2 − G (x)) e−G(x) , x ∈ R.

(1.1.598)

and

1.1.298

Zero-Truncated Poisson Exponentiated Gamma (ZTPEG)

The cd f and pd f of ZTPEG are given, respectively, by n o θ 1 − exp −α [1 − (x + 1) e−x ] F (x; α, θ) = , 1 − e−α and

f (x; α, θ) =

αθxe−x [1 − (x + 1) e−x ]

θ−1

x ≥ 0,

n o θ exp −α [1 − (x + 1) e−x]

1 − e−α

(1.1.599)

,

(1.1.600)

x > 0, where α > 0, θ > 0 are parameters. θ

Remark 1.220. Taking G (x) = [1 − (x + 1) e−x ] , x ≥ 0, the cd f (1.1.599) can be expressed as F (x; α, θ) = which was discussed in Remark 1.39.

1.1.299

1 − exp {−αG (x)} , x ≥ 0, 1 − e−α

Power Gompertz (PG)

The cd f and pd f of PG are given, respectively, by  θ  − αβ eβx −1

F (x; α, β, θ) = 1 − e

,

x ≥ 0,

(1.1.601)

and  θ  α βx θ−1 βxθ − β e −1

f (x; α, β, θ) = αθx

e

e

, x > 0,

(1.1.602)

where α > 0, β > 0, θ > 0 are parameters. θ

Remark 1.221. Taking G (x) = 1 − e−βx , x ≥ 0, the cd f (1.1.601) can be written as − αβ

F (x; α, β, θ) = 1 − e



G(x) G(x)



, x ≥ 0,

which is a special case of the cd f mentioned in Remark 1.6.

Introduction

1.1.300

145

Transmuted New Weibull Pareto (TNWP)

The cd f and pd f of TNWP are given, respectively, by   β β −δ( θx ) −δ( θx ) F (x; β, δ, λ, θ) = 1 − e 1 − λ + λe ,

x ≥ 0,

(1.1.603)

and

  β δβ β−1 −δ( θx )β −δ( θx ) 1 − λ + 2λe , x > 0, f (x; β, δ, λ, θ) = β x e θ

(1.1.604)

where β > 0, δ > 0, λ ∈ [−1, 1] and θ > 0 are parameters. x β

Remarks 1.222. (a) Taking G (x) = 1 − e−δ( θ ) , x ≥ 0, the cd f (1.1.603) can be written as F (x; β, δ, λ, θ) = (1 + λ) G (x) − λ [G (x)]2 , x ≥ 0,

which has been characterized in Hamedani (2019). That being said we will provide certain characterizations of the TNWP in the following sections. (b) for λ = 0 and λ = 1, the cd f (1.1.603) will reduce to the cd f 0 s of Weibull distributions.

1.1.301

Lomax Weibull (LoW)

The cd f and pd f of LoW are given, respectively, by "

(

1 − e−(x/σ)µ F (x; α, β, σ, µ) = k 1 − 1 + β

)−α #

, x ≥ 0,

(1.1.605)

and kαµ  x µ−1 −(x/σ)µ e × σβ σ ( )−α−1 1 − e−(x/σ)µ 1+ , x > 0, β

f (x; α, β, σ, µ) =

(1.1.606)

h  α i−1 β where α, β, σ, µ are all positive parameters and k = 1 − β+1 is the normalizing constant.  µ −1 Remark 1.223. The authors have k = 1 − e−1/σ , which is clearly incorrect.

G. G. Hamedani

146

1.1.302

Odd Log-Logistic Geometric-G (OLLG-G)

The cd f and pd f of OLLG-G are given, respectively, by F (x; α, p, τ) = and

Gα (x; τ) α

Gα (x; τ) + (1 − p) G (x; τ)

, x ∈ R,

d F (x; α, p, τ), x ∈ R, dx where α > 0, p ∈ (0, 1), τ > 0 are parameters. f (x; α, p, τ) =

(1.1.607)

(1.1.608)

Remark 1.224. The cd f (1.1.607) has been characterized in Hamedani (2016).

1.1.303

Marshall-Olkin Extended Quasi Lindley (MOEQL)

The cd f and pd f of MOEQL are given, respectively, by   −θx 1 − (β+1+θx) β+1 e   , x ≥ 0, F (x; α, β, θ) = (β+1+θx) −θx 1 − (1 − α) e β+1

(1.1.609)

and

d F (x; α, β, θ), x > 0, (1.1.610) dx where α > 0, β > −1, θ > 0 are parameters.   −θx , x ≥ 0, the cd f (1.1.609) can be exe Remark 1.225. Taking G (x) = 1 − (β+1+θx) β+1 pressed as f (x; α, β, θ) =

G (x) , x ≥ 0, 1 − θ (1 − α) G (x) which is a special case of the cd f discussed in Remark 1.38. F (x; α, β, θ) =

1.1.304

Poisson Rayleigh Log-Logistic (PRLL)

The cd f and pd f of PRLL are given, respectively, by F (x; α, λ) = and

  2α −λ 1−e−x

1−e 1 − e−λ

f (x; α, λ) =

, x ≥ 0,

d F (x; α, λ), x > 0, dx

(1.1.611)

(1.1.612)

where α > 0, λ > 0 are parameters. Remark 1.226. The cd f (1.1.611) is not new. It has been proposed by Almamy (2019). Please see the cd f (1.1.111).

Introduction

1.1.305

147

Generalized Marshall-Olkin Extended Burr XII (GMOBXII)

The cd f and pd f of GMOBXII are given, respectively, by F (x; α, β, λ, θ) = 1 −

"

and

α 1 + xλ


−β #θ

1 − α 1 + xλ


−β

, x ≥ 0,

(1.1.613)

d F (x; α, β, λ, θ), x > 0, (1.1.614) dx where α, β, λ, θ are all positive parameters and α = 1 − α.
−β Remark 1.227. Taking G (x) = 1 − 1 + xλ , x ≥ 0, the cd f (1.1.613) can be written as f (x; α, β, λ, θ) =



αG(x) F (x; α, β, λ, θ) = 1 − 1 − αG (x)

θ

, x ≥ 0,

which was proposed by Handique et al. (2019). Clearly, the cd f (1.1.613) is a special case of the above cd f . Please see Remark 1.38 as well.

1.1.306

Type I New Heavy Tailed Weibull (TINHT-W)

The cd f and pd f of TINHT-W are given, respectively, by

and

α

e−γx F (x; α, θ, γ) = 1 − 1 − (1 − θ) (1 − e−γxα ) 

θ

, x ≥ 0,

d F (x; α, θ, γ), x > 0, dx

f (x; α, θ, γ) =

(1.1.615)

(1.1.616)

where α, θ, γ are all positive parameters. α

Remark 1.228. Taking G (x) = 1 − e−γx , x ≥ 0, the cd f (1.1.615) can be written as F (x; α, θ, γ) = 1 −



G (x) 1 − (1 − θ) G (x)

θ

, x ≥ 0,

which is the same as the cd f given in Remark 1.193 for θ = 1γ .

1.1.307

Arcsine Weibull (AS-W)

The cd f and pd f of AS-W are given, respectively, by F (x; α, γ) = and

  2 α arcsin 1 − e−γx , π

x ≥ 0,

(1.1.617)

G. G. Hamedani

148

f (x; α, γ) =

d F (x; α, γ), x > 0, dx

(1.1.618)

where α, γ are positive parameters. α

Remark 1.229. Taking G (x) = 1 − e−γx , x ≥ 0, the cd f (1.1.617) can be written as 2 arcsin (G (x)), x ≥ 0, π which has been characterized in our previous work. F (x; α, γ) =

1.1.308

Truncated Inverted Kumaraswamy Generated (TIK-G)

The cd f and pd f of TIK-G are given, respectively, by F (x; a, b, η) = and

f (x; a, b, η) =

1 (1 − 2−a )b



b 1 − (1 + G (x; η))−a ,

abg (x; η) (1 + G (x; η))−a−1  (1 − 2−a )b

x ∈ R,

(1.1.619)

b−1

, x ∈ R, (1.1.620)

1 − (1 + G (x; η))−a

where a > 0, b > 0 are parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η).

1.1.309

Lomax Gompertz-Makeham (LOGOMA)

The cd f and pd f of LOGOMA are given, respectively, by

and

io−a n h  −θx− αβ (eβx −1) , x ≥ 0, F (x; α, β, θ, a, b) = 1 − ba b − log 1 − 1 − e

(1.1.621)

f (x; α, β, θ, a, b) =
α βx aba θ + αeβx e−θx− β (e −1) h  in h  ioa+1 , x > 0, −θx− αβ (eβx −1) −θx− αβ (eβx−1) 1− 1−e b − log 1 − 1 − e

where α, β, θ, a, b are all positive parameters.

(1.1.622)

Introduction

1.1.310

149

Exponential Transmuted Fréchet (ETF)

The cd f and pd f of ETF are given, respectively, by n h  ioθ α α F (x; α, β, θ, λ) = 1 − 1 − e−(β/x) 1 + λ 1 − e−(β/x) ,

(1.1.623)

x ≥ 0, and

  α α θαβαx−(α+1)e−(β/x) 1 + λ − 2λe−(β/x) f (x; α, β, θ, λ) =  , α α
 1−θ 1 − e−(β/x) 1 + λ 1 − e−(β/x)

(1.1.624)

where α, β, θ all positive and λ ∈ [−1, 1] are parameters.

1.1.311

Marshall-Olkin Extended Weibull Exponential (MOEWE)

The cd f and pd f of MOEWE are given, respectively, by h
β i 1 − exp −α eλx − 1 h F (x; α, β, θ, λ) =
β i , 1 − (1 − θ) exp −α eλx − 1

(1.1.625)

x ≥ 0, and

d F (x; α, β, θ, λ), x > 0, (1.1.626) dx where α, β, θ, λ are all positive parameters. h
β i Remark 1.230. Taking G (x) = 1 − exp −α eλx − 1 , x ≥ 0, the cd f (1.1.625) can be written as f (x; α, β, θ, λ) =

F (x; α, β, θ, λ) =

G (x) , x ≥ 0, 1 − (1 − θ) G (x)

which is a special case of the cd f discussed in Remark 1.38.

1.1.312

Extended Log-Logistic (ELL)

The cd f and pd f of ELL are given, respectively, by "

xβ F (x; α, β, a, b) = 1 − 1 − β α + xβ

!a #b

, x ≥ 0,

(1.1.627)

and d F (x; α, β, a, b), x > 0, dx where α, β, a, b are all positive parameters. f (x; α, β, a, b) =

(1.1.628)

G. G. Hamedani

150 Remark 1.231. Taking G (x) =

xβ , αβ +xβ

x ≥ 0, the cd f (1.1.627) can be written as b

F (x; α, β, a, b) = [1 − (1 − G (x))a ] , x ≥ 0,

which has been characterized in Hamedani (2018).

1.1.313

Log-Odd Normal Generalized (LONG)

The cd f and pd f of LONG are given, respectively, by    1 1 G (x) √ ln F (x; λ) = 1 + erf , x ∈ R, 2 λ 2 G (x) and   g (x) 1 2 G (x) f (x; λ) = √ exp − 2 ln , x ∈ R, 2λ G (x) λ 2G (x) G (x)

(1.1.629)

(1.1.630)

where λ > 0 is a parameter.

1.1.314

Marshall-Olkin Generalized Burr XII (MOGBXII)

The cd f and pd f of MOGBXII are given, respectively, by F (x; α, β, δ, a) = and

1 − (1 + xα )−aβ

1 − (1 − δ) (1 + xα )−aβ

, x ≥ 0,

d F (x; α, β, δ, a), x > 0, dx where α, β, δ, a are all positive parameters. f (x; α, β, δ, a) =

(1.1.631)

(1.1.632)

Remark 1.232. Taking G (x) = 1 − (1 + xα )−aβ , x ≥ 0, the cd f (1.1.631) can be written as G (x) , x ≥ 0, 1 − (1 − δ) G (x) which has been discussed in Remark 1.44. F (x; α, β, δ, a) =

1.1.315

Generalized Inverse Lindley (GIL)

The cd f and pd f of GIL are given, respectively, by F (x; α, β, λ) =

β + λ + βλx−α −βx−α e , x ≥ 0, β+λ

(1.1.633)

and αβ2 x−α−1 (1 + λx−α ) −βx−α e , x > 0, β+λ where α, β, λ are all positive parameters. f (x; α, β, λ) =

(1.1.634)

Introduction

151

Remark 1.233. Alkarni continues to define GILPS with the following cd f i  h −α −βx−α e C θ 1 − β+λ+βλx β+λ , x ≥ 0, F (x; θ, α, β, λ) = C (θ) which can be written as F (x; θ, α, β, λ) = −α

C (θG (x)) , x ≥ 0, C (θ)

−α

e−βx , x ≥ 0. The last cd f has been characterized in where G (x) = 1 − β+λ+βλx β+λ Hamedani (2018a). Here we will characterize the GIL distribution.

1.1.316

Normal-C (N-C)

The cd f and pd f of N-C are given, respectively, by   2G (x) − 1 F (x) = Φ , x ∈ R, G (x) G (x)

(1.1.635)

and

2G (x) − 1 f (x) = φ G (x) G (x) 



1 − 2G (x) G (x) G (x)2 G (x)2

g (x) , x ∈ R,

(1.1.636)

where φ (x) , Φ (x) are pd f and cd f of standard normal and G (x) is a continuous baseline cd f .

1.1.317

Extended Power Lindley-G (EPL-G)

The cd f and pd f of EPL-G are given, respectively, by α h
α io e−θ(− log G(x;φ)) n F (x; α, θ, φ) = 1 − , x ∈ R, 1 + θ 1 + − log G (x; φ) 1+θ

(1.1.637)

and

n
α o αθ2 g (x; φ) 1 + − log G(x; φ) f (x; α, θ, φ) =  
1−α × 1 + θG (x; φ) − log G(x; φ) α

e−θ(− logG(x;φ)) , x ∈ R,

(1.1.638)

where α, θ, φ are all positive parameters and G (x; φ) is a baseline cd f with the corresponding pd f g (x; φ).

G. G. Hamedani

152

1.1.318

Type II Topp-Leone Inverted Kumaraswamy (TIITLIK)

The cd f and pd f of TIITLIK are given, respectively, by  α  2b 1 − 1 − (1 + x)−a × F (x; α, a, b, λ) = 1 −  n  b o2  , 1 + λ − λ 1 − (1 + x)−a

(1.1.639)

x ≥ 0, and

d F (x; α, a, b, λ), x > 0, (1.1.640) dx where α, a, b all positive and λ ∈ [−1, 1] are parameters.  b Remark 1.234. Taking G (x) = 1 − (1 + x)−a , x ≥ 0, the cd f (1.1.639) can be written as n oα F (x; α, a, b, λ) = 1 − 1 − G (x)2 [1 + λ − λG (x)]2 , x ≥ 0, f (x; α, a, b, λ) =

which is a special case of the cd f discussed in Remark 1.42.

1.1.319

New Power Topp-Leone Generated (NPTL-G)

The cd f and pd f of NPTL-G are given, respectively, by     α 1 1 αβ 1− G(x;λ) β 1− G(x;λ) F (x; α, β, λ) = e 2−e , x ∈ R,

(1.1.641)

x ≥ 0, and

d F (x; α, β, λ), x ∈ R, dx where α, β, λ are all positive parameters and G (x; λ) is a baseline cd f . f (x; α, β, λ) =

  1 β 1− G(x;λ)

Remark 1.235. Taking K (x) = e

(1.1.642)

, x ∈ R, the cd f (1.1.641) can be written as

F (x; α, β, λ) = K (x)α [2 − K (x)]α , x ∈ R,

which has been characterized in Hamedani (2017).

1.1.320

Modified Beta Gompertz (MBG)

The cd f and pd f of MBG are given, respectively, by − λ eαx −1) c 1−e α (

1 F (x; α, λ, a, b, c) = B (a, b) and

Z

0

!

λ αx 1−(1−c) 1−e− α (e −1)

!

t a−1 (1 − t)b−1 dt, x ≥ 0,

(1.1.643)

Introduction

153

d F (x; α, λ, a, b, c), dx where α, λ, a, b, c are all positive parameters. f (x; α, λ, a, b, c) =

Remark 1.236. Taking Q (x) = ten as

  λ αx c 1−e− α (e −1)  , λ αx 1−(1−c) 1−e− α (e −1)

F (x; α, λ, a, b, c) =

1 B (a, b)

Z Q(x) 0

x > 0,

(1.1.644)

x ≥ 0, the cd f (1.1.643) can be writ-

t a−1 (1 − t)b−1 dt, x ≥ 0,

which has been discussed in Remark 1.15.

1.1.321

Exponentiated Weibull-Exponentiated Weibull (EW-EW)

The cd f and pd f of EW-EW are given, respectively, by 

and

−

F (x; α, β, θ, a, b, c) = 1 − 1 − e



− βc

  a α θ −( x ) b ln 1−e

 ,

i x a −1 ac αθ  x a−1 −( x )a h e b 1 − e−( b ) b β b   a α  iα−1 − − c ln 1−e−( xb ) x a c h β × − ln 1 − e−( b ) e β     a α θ−1 − x − − βc ln 1−e ( b )  , x > 0, × 1 − e

(1.1.645)

f (x; α, β, θ, a, b, c) =

(1.1.646)

where α, β, θ, a, b, c are all positive parameters.

1.1.322

Nadarajah Haghighi Topp Leaone-G (NHTL-G)

The cd f and pd f of NHTL-G are given, respectively, by F (x; α, λ, θ, φ) = and

n o θ α 1− 1+λ(1−G(x;φ)2 )

1−e

1 − e−λ

, x ∈ R,

(1.1.647)

G. G. Hamedani

154

 θ−1 × f (x; α, λ, θ, φ) = 2αλθg (x; φ) G (x; φ) 1 − G (x; φ)2   θ α−1 2 1 + λ 1 − G (x; φ) o , x ∈ R, n
−1+ 1+λ(1−G(x;φ)2 )θ α −λ 1−e e

(1.1.648)

where α, λ, θ, φ are all positive parameters. Remark 1.237. The factor

1.1.323

1 1−e−λ

is missing in the original paper.

Weibull Generalized Burr XII (WGBXII)

The cd f and pd f of WGBXII are given, respectively, by  h iβ  a θb F (x; β, θ, a, b) = 1 − exp − (1 + x ) − 1 ,

(1.1.649)

x ≥ 0, and

βθabxa−1 (1 + xa )θb−1 f (x; β, θ, a, b) = h i1−β × θb a (1 + x ) − 1  h iβ  a θb exp − (1 + x ) − 1 , x > 0,

(1.1.650)

where β, θ, a, b are all positive parameters.

1.1.324

Extended Poisson Fréchet (EPFr)

The cd f and pd f of EPFr are given, respectively, by  ( θ  )2   exp −αβ x−β    + 1 , x ≥ 0, F (x; α, β, θ) = − exp − 1 − exp −αβ x−β 

and

(1.1.651)

d F (x; α, β, θ), x > 0, (1.1.652) dx where α, β, θ are all positive parameters.   Remark 1.238. Taking G (x) = exp −αβ x−β , x ≥ 0, the cd f (1.1.651) can be expressed as f (x; α, β, θ) =

"

G (x) F (x; α, β, θ) = 1 − exp − G (x) which was discussed in Remark 1.2.



2 !#θ

, x ≥ 0,

Introduction

1.1.325

155

Poisson Burr X -Fréchet (PBX-Fr)

The cd f and pd f of PBX-Fr are given, respectively, by

F (x; λ, β, θ) =

"  #  h i−2 θ −β x 1 − exp − 1 − exp − e − 1 1 − e−λ

, x ≥ 0,

(1.1.653)

and f (x; λ, β, θ) =

d F (x; λ, β, θ), x > 0, dx

(1.1.654)

where λ, β, θ are all positive parameters.  h i−2  x−β Remark 1.239. Taking G (x) = 1 − exp − e − 1 , x ≥ 0, the cd f (1.1.653) can be

expressed as

θ

1 − e−λG(x) , x ≥ 0, F (x; λ, β, θ) = 1 − e−λ which was discussed in Remark 1.40.

1.1.326

Odd Log-Logistic Lindley-G (OLLLi-G)

The cd f and pd f of OLLLi-G are given, respectively, by F (x; α, θ, φ)    θG(x;φ) α − θG(x;φ) 1 − 1 + (θ+1)G(x;φ) e G(x;φ) =   θG(x;φ) α   θG(x;φ) α , x ∈ R, (1.1.655) − G(x;φ) − θG(x;φ) θG(x;φ) 1 − 1 + (θ+1)G(x;φ) e + 1 + (θ+1)G(x;φ) e G(x;φ) and d F (x; α, θ, φ), x ∈ R, (1.1.656) dx where α, θ, φ are all positive parameters.  θG(x;φ)  − θG(x;φ) Remark 1.240. Taking K (x) = 1 − 1 + (θ+1)G(x;φ) e G(x;φ) , x ∈ R, the cd f (1.1.655) can be expressed as f (x; α, θ, φ) =

F (x; α, θ, φ) = which was discussed in Remark 1.39.

K (x)α , x ∈ R, K (x)α + K (x)α

G. G. Hamedani

156

1.1.327

Exponentiated Generalized Power Lindley (EG-PL)

The cd f and pd f of EG-PL are given, respectively, by F (x; β, λ, a, b) =

(

1−

"

λxβ 1+ 1+λ

!

−λxβ

e

#a )b

,

(1.1.657)

x ≥ 0, and " ! #a−1  β abλ2 β−1  β λx β f (x; β, λ, a, b) = x 1 + xβ e−λx 1+ e−λx × 1+λ 1+λ ( " ! #a )b−1 λxβ β 1− 1+ e−λx , x > 0. (1.1.658) 1+λ x > 0, where β, λ, a, b are all positive parameters.

1.1.328

Generalized Odd Half-Cauchy-G (GOHC-G)

The cd f and pd f of GOHC-G are given, respectively, by   2 G (x; φ)α F (x; α, φ) = arctan , π 1 − G (x; φ)α and

x ∈ R,

2α (x; φ)G (x; φ)α−1 f (x; α, φ) = n  2 o , x ∈ R, π G (x; φ)2α + 1 − G (x; φ)α

(1.1.659)

(1.1.660)

where α > 0 is a parameter and G (x; φ) is a baseline cd f with the corresponding pd f g (x; φ) which may depend on the parameter vector φ.

1.1.329

Generalized Kumaraswamy-G (GK-G)

The cd f and pd f of GK-G are given, respectively, by F (x; α, a, b, φ) =

1 − [1 − αG (x; ϕ)a ] 1 − (1 − α)b

and

f (x; α, a, b, φ) =

αabg (x; ϕ) G (x; ϕ)a−1 1 − (1 − α)

b

b

,

[1 − αG (x; ϕ)a ]

x ∈ R,

b−1

, x ∈ R,

(1.1.661)

(1.1.662)

where a > 0, b > 0 and α ∈ (0, 1] are parameters and G (x; ϕ) is a baseline cd f with the corresponding pd f g (x; ϕ) which may depend on the parameter vector ϕ.

Introduction

1.1.330

157

Alpha Power Inverted Exponential (APIE)

The cd f and pd f of APIE are given, respectively, by ( exp(−λ/x) α −1 , if α > 0, α 6= 1 α−1 F (x; α, λ) = , exp(−λ/x), if α = 1

x ≥ 0,

(1.1.663)

and f (x; α, λ) =

d F (x; α, λ), x > 0, dx

(1.1.664)

where a and λ > 0 are parameters. Remark 1.241. Taking G (x) = exp (−λ/x) , x ≥ 0, the cd f (1.1.663) can be written as  αG(x) −1 α−1 , i f α > 0, α 6= 1 F (x; α, λ) = G(x) , x ≥ 0, , if α = 1

which has been considered in Remark 1.143.

1.1.331

Alpha Power Inverse Weibull (APIW)

The cd f and pd f of APIW are given, respectively, by ( exp(−λ/xβ ) α −1 , if α > 0, α 6= 1 α−1 F (x; α, β, λ) = , exp(−λ/xβ ), if α = 1

x ≥ 0,

(1.1.665)

and d F (x; α, β, λ), x > 0, (1.1.666) dx where a and β > 0, λ > 0 are parameters.
Remark 1.242. Taking G (x) = exp −λ/xβ , x ≥ 0, the cd f (1.1.665) can be written as  αG(x) −1 α−1 , i f α > 0, α 6= 1 F (x; α, β, λ) = G(x) , x ≥ 0, , if α = 1 f (x; α, β, λ) =

which has been considered in Remark 1.143. Note that the cd f (1.1.665) is a slight extension of the cd f (1.1.663).

1.1.332

Kumarswamy Log-Logistic Weibull (KLLoGW)

The cd f and pd f of KLLoGW are given, respectively, by

and

n h i o β a b F (x; α, β, a, b, c) = 1 − 1 − 1 − (1 + xc )−1 e−αx ,

x ≥ 0,

(1.1.667)

G. G. Hamedani

158

d F (x; α, β, a, b), x > 0, dx where a, β, a, b, c are all positive parameters. f (x; α, β, λ) =

(1.1.668)

β

Remark 1.243. Taking K (x) = 1 − (1 + xc )−1 e−αx , x ≥ 0, the cd f (1.1.667) will be a special case of the cd f considered in the Remarks 1.12 and 1.21.

1.1.333

Mrashall-Olkin Kappa (MOK)

The cd f and pd f of MOK are given, respectively, by

and

  −1 1/α αθ −αθ F (x; α, β, θ, δ) = δ 1 + αβ x + (1 − δ) , x ≥ 0,
1 −1 δαβαθ x−αθ−1 1 + αβαθ x−αθ α f (x; α, β, θ, δ) = h i2 , x > 0, 1/α αθ −αθ δ(1 + αβ x ) + (1 − δ)

(1.1.669)

(1.1.670)

where a, β, θ, δ are all positive parameters.

1.1.334

Exponentiated Gumbel Exponential (EGuE)

The cd f and pd f of EGuE are given, respectively, by    −1/σα θx F (x; α, θ, σ, µ) = 1 − 1 − exp −B e − 1 , x ≥ 0,

(1.1.671)

and

  − σ1 −1 −1/σ αθB θx  θx θx f (x; α, θ, σ, µ) = e e −1 × exp −B e − 1 σ    −1/σα−1 θx × 1 − exp −B e − 1 , (1.1.672) x > 0, where a > 0, θ > 0, σ > 0, µ ∈ R are parameters and B = eµ/σ .

1.1.335

Type II Topp-Leone Power Lomax (TIITLPL)

The cd f and pd f of TIITLPL are given, respectively, by !−α #2 θ  x F (x; α, β, λ, θ) = 1 − 1 − 1 − 1 + , x ≥ 0,   λ  

and

"

β

(1.1.673)

Introduction

159

d F (x; α, β, λ, θ), x > 0, dx where α, β, λ, θ are all positive parameters. f (x; α, β, λ, θ) =

(1.1.674)

Remark 1.244. A more general version of the cd f (1.1.673) is discussed in Remark 1.100.

1.1.336

XGamma-G (XG-G)

The cd f and pd f of XG-G are given, respectively, by θ2 F (x; θ, η) = 1+θ and

Z − log G(x;η) 

 θ 2 −θt 1 + t e dt, x ∈ R, 2

0

  2 θ 1 2 θ−1 f (x; θ, η) = g (x; η) G (x; η) θ + θ log G(x; η) , x ∈ R, 1+θ 2

(1.1.675)

(1.1.676)

where θ > 0 is a parameter and G (x; η) is a baseline cd f with the corresponding pd f g (x; η), which may depend on the parameter vector η.

1.1.337

Weibull-Inverse Lomax (WIL)

The cd f and pd f of WIL are given, respectively, by (   )  γ λ −β F (x; α, β, γ, λ) = 1 − exp α 1 − 1 + , x ≥ 0, x

(1.1.677)

and

d F (x; α, β, γ, λ), x > 0, dx where α, β, γ, λ are all positive parameters. f (x; α, β, γ, λ) =

(1.1.678)

Remarks 1.245. (a) The equation (1.1.677) is not a cd f unless it is assumed that β is chosen such that (−1)β = −1. (b) The cd f (1.1.677) is a special case of the cd f (1.1.435).

1.1.338

Kumaraswamy Alpha Power Inverted Exponential (KAPIE)

The cd f and pd f of KAPIE are given, respectively, by (

F (x; α, λ, ψ, b) = 1 − 1 − and

−λ/x

αe − 1 α−1

!ψ)b

, x ≥ 0,

d F (x; α, λ, ψ, b), x > 0, dx where α > 0 (α 6= 1), λ > 0, ψ > 0 and b > 0 are parameters. f (x; α, λ, ψ, b) =

(1.1.679)

(1.1.680)

160 Remark 1.246. Taking G (x) =



G. G. Hamedani ψ −λ/x αe −1 , x ≥ 0, the cd f (1.1.679) can be written as α−1

F (x; α, λ, ψ, b) = 1 − {1 − G (x)}b , x ≥ 0,

which has been characterized in our previous work.

1.1.339

Odd Generalized Exponential-Exponential (OGE-E)

The cd f and pd f of OGE-E are given, respectively, by  α γx F (x; α, λ, γ) = 1 − e−λ(e −1) , x ≥ 0,

(1.1.681)

and

f (x; α, λ, γ) =

d F (x; α, λ, γ), x > 0, dx

(1.1.682)

where α, λ, γ are all positive parameters. Remark 1.247. Hassan and Elgarhy (2018) proposed the following distribution "   #a −α

F (x; α, β, a, η) = 1 − e

K(x;η) 1−K(x;η)

β

, x ∈ R.

Taking K (x; η) = 1−e−γx, x ≥ 0, F (x; α, β, a, η) reduces to the cd f (1.1.681) and hence provides a more general distribution. The cd f F (x; α, β, a, η) has been characterized in Hamedani (2019).

1.1.340

Topp-Leone-Lomax (TLLx)

The cd f and pd f of TLLx are given, respectively, by " −2α2 #α1  x , x ≥ 0, F (x; α1 , α2 , α3 ) = 1 − 1 + α3

(1.1.683)

and d F (x; α1 , α2 , α3 ) , x > 0, dx where α1 , α2 , α3 are all positive parameters. f (x; α1 , α2 , α3 ) =

Remark 1.248. Gómez et al. (2014) proposed the following distribution h iα F (x; α, θ, λ) = 1 − (1 + λx)−θ , x ≥ 0,

(1.1.684)

which is the same as the cd f (1.1.683). Furthermore, a generalization of Gómez et al. was introduced by Mead (2015) which has been characterized in Hamedani and Maadooliat (2017).

Introduction

1.1.341

161

Beta Generalized Exponentiated Fréchet (BGEF)

The cd f and pd f of BGEF are given, respectively, by

F (x; α, β, θ, λ, a, b) =

1 B (a, b)



Z 1− 1−e−x−α 0



θλ β

t a−1 (1 − t)b−1 dt, x ≥ 0,

(1.1.685)

and d F (x; α, β, θ, λ, a, b), x > 0, (1.1.686) dx where α, β, θ, λ, a, b are all positive parameters.    β −α θλ −x Remark 1.249. Taking Q (x) = 1 − 1 − e in (1.1.685), we arrive at f (x; α, β, θ, λ, a, b) =

Q(x) 1 t a−1 (1 − t)b−1 dt, x ≥ 0, B (a, b) 0 which has been introduced before and characterized in Hamedani (2016).

F (x; α, β, θ, λ, a, b) =

1.1.342

Z

Transmuted Generalized Inverted Exponential (TGIE)

The cd f and pd f of TGIE are given, respectively, by

and

 φ    φ   F (x; θ, λ, φ) = 1 − 1 − e−θ/x 1 + λ 1 − e−θ/x , x ≥ 0,

d F (x; θ, λ, φ), x > 0, dx where θ > 0, λ (|λ| ≤ 1), φ > 0 are parameters. f (x; θ, λ, φ) =

(1.1.687)

(1.1.688)

Remark 1.250. The cd f discussed in Remark 1.19 is more general than the cd f (1.1.687).

1.1.343

Transmuted Generalized Power Weibull (TGPW)

The cd f and pd f of TGPW are given, respectively, by

and

 h i  h i 1−(1+xα )β 1−(1+xα )β F (x; α, β, λ) = 1 − e 1 + λe , x ≥ 0,

d F (x; α, β, λ), x > 0, dx where α > 0, β > 0 and λ (|λ| ≤ 1) are parameters. f (x; α, β, λ) =

(1.1.689)

(1.1.690)

Remark 1.251. Please see Remark 1.28. The cd f (1.1.689) is also similar to the cd f (1.1.473).

162

1.1.344

G. G. Hamedani

Size Biased Gamma Lindley (SBGaL)

The cd f and pd f of SBGaL are given, respectively, by  2  θ (β + βθ − θ) 2 F (x; β, θ) = 1 − x + θx + 1 e−θx , x ≥ 0, 2β (1 + θ) − θ

(1.1.691)

and

f (x; β, θ) =

d F (x; β, θ), x > 0, dx

(1.1.692)

where β > 0 and θ > 0 are parameters. Remark 1.252. The cd f (1.1.245) is more general than the cd f (1.1.691). Please see the cd f s (1.1.519) and (1.1.525) as well.

1.1.345

Skew t-Distribution of Three Degrees of Freedom (St-DTDF)

The cd f and pd f of St-DTDF are given, respectively, by F (x; a) = and

Z x

−∞

" √ 6 3 2 1+ f (x; a) = 2 π (3 + x ) π

f (u; a)du, x ∈ R,

(1.1.693)

!# √  ax 3ax arctan √ + , 3 + a2 x2 3 

(1.1.694)

x ∈ R, where a ∈ R\ {0} is a parameter.

1.1.346

Logarithm Transformed Lomax (LTL)

The cd f and pd f of LTL are given, respectively, by   λ β 1−( λ+x ) log 1 + α + α F (x; α, β, λ) = 1 − , log(α)

x ≥ 0,

(1.1.695)

and λ

f (x; α, β, λ) =

β

βλβ (λ + x)−(β+1) α1−( λ+x ) 1+α+α

λ 1−( λ+x )

x > 0, where α > 0 (α 6= 1) , β > 0, λ > 0 are parameters.

β

,

(1.1.696)

Introduction

1.1.347

163

Exponentiated Generalized Power Function (EGPF)

The cd f and pd f of EGPF are given, respectively, by    x θ α β F (x; α, β, λ, θ) = 1 − 1 − , 0 ≤ x ≤ λ, λ

and

(1.1.697)

d F (x; α, β, λ, θ), 0 < x < λ, (1.1.698) dx where α, β, λ and θ are all positive parameters.
θ Remark 1.253. Taking G (x) = λx , 0 ≤ x ≤ λ, the cd f (1.1.697) can be written as f (x; α, β, λ, θ) =

 β F (x; α, β, λ, θ) = 1 − (1 − G (x))α ,

which is a special case of the cd f s discussed in Remarks 1.109 and 1.114.

1.1.348

Gompertz Length Biased Exponential (Go-LBE)

The cd f and pd f of Go-LBE are given, respectively, by F (x; β, θ, γ) = 1 − e

θ γ



1−

h

 i−γ  1+ βx e−x/β

, x ≥ 0,

(1.1.699)

and θ γ

f (x; β, θ, γ) =



1−

h

 i−γ  1+ βx e−x/β

θxe−x/β e h  iγ+1 β2 1 + βx e−x/β

, x > 0,

(1.1.700)

where β, θ and γ are all positive parameters.

1.1.349

Gumbel-Burr XII (GUBXII)

The cd f and pd f of GUBXII are given, respectively, by (

 −1/α)  x s λ F (x; α, λ, ε, c, s) = exp −eε/α 1 + −1 , x ≥ 0, c

(1.1.701)

and d F (x; α, λ, ε, c, s), x > 0, dx where α, λ, ε, c, s are all positive parameters. f (x; α, λ, ε, c, s) =

(1.1.702)

Remark 1.254. The characterizations of the cd f (1.1.701) are similar to those of the cd f (1.1.435).

G. G. Hamedani

164

1.1.350

Type II Power Topp-Leone Generated (TIIPTL-G)

The cd f and pd f of TIIPTL-G are given, respectively, by

and

n oα F (x; α, β, η) = 1 − [1 − G (x; η)]αβ 2 − [1 − G (x; η)]β , x ∈ R,

(1.1.703)

n oα−1 f (x; α, β, η) = 2αβg (x; η) [1 − G (x; η)]αβ−1 2 − [1 − G (x; η)]β n o × 1 − [1 − G (x; η)]β , (1.1.704) x ∈ R, where α, β are positive parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η) which may depend on the parameter vector η.

1.1.351

Beta Kumaraswamy Marshall-Olkin-G (BKwMO-G)

The cd f and pd f of BKwMO-G are given, respectively, by

F (x; α, a, b, m, n) =

1 × B (m, n) h

Z 1− 1− 0



a ib G(x) 1−αG(x)

t m−1 (1 − t)n−1 dt, x ∈ R,

(1.1.705)

and d F (x; α, a, b, m, n), x ∈ R, (1.1.706) dx where α, a, b, m, n are all positive parameters, α = 1 − α and G (x) is a baseline cd f with the corresponding pd f g (x). h   a ib G(x) Remark 1.255. Taking Q (x) = 1 − 1 − 1−αG(x) , x ∈ R, the cd f (1.1.705) can be expressed as f (x; α, a, b, m, n) =

F (x; α, a, b, m, n) =

1 B (m, n)

which has been discussed in Remark 1.15.

Z Q(x) 0

t m−1 (1 − t)n−1 dt, x ∈ R,

Introduction

1.1.352

165

Beta Generalized Marshall-Olkin Kumarswamy-G (BGMOKw-G)

The cd f and pd f of BGMOKw-G are given, respectively, by

F (x; α, θ, a, b, m, n) =

1 × B (m, n)

Z 1−



α(1−G(x)a )b 1−α(1−G(x)a )b

θ

0

t m−1 (1 − t)n−1 dt, x ∈ R,

(1.1.707)

and d F (x; α, θ, a, b, m,n), x ∈ R, (1.1.708) dx where α, θ, a, b, m, n are all positive parameters, α = 1 − α and G (x) is a baseline cd f with the corresponding pd f g (x). f (x; α, θ, a, b, m, n) =

Remark 1.256. Taking Q (x) = 1 −

pressed as

F (x; α, a, b, m, n) =

h

b

α(1−G(x)a ) 1−α(1−G(x)a )b

1 B (m, n)

Z Q(x) 0

iθ

, x ∈ R, the cd f (1.1.707) can be ex-

t m−1 (1 − t)n−1 dt, x ∈ R,

which has been discussed in Remark 1.15.

1.1.353

Beta Marshall-Olkin Kumarswamy-G (BMOKw-G)

The cd f and pd f of BMOKw-G are given, respectively, by

1 F (x; l, a, b, m, n) = B (m, n)

Z 1−



1−(1−G(x)a )b 1−l(1−G(x)a )b

0



t m−1 (1 − t)n−1 dt, x ∈ R,

(1.1.709)

and f (x; l, a, b, m, n) =

d F (x; l, a, b, m, n), x ∈ R, dx

(1.1.710)

where l, a, b, m, n are all positive parameters, l = 1 − l and G (x) is a baseline cd f with the corresponding pd f g (x). Remark 1.257. The cd f (1.1.707) of the same authors is a slight extension of their cd f (1.1.709).

G. G. Hamedani

166

1.1.354

Zografos-Balakrishnan Burr XII (ZBBXII)

The cd f and pd f of ZBBXII are given, respectively, by F (x; α, β, a) =

Z − log(1+xα )−β

1 Γ (a)

0

t a−1 e−t dt, x ≥ 0,

(1.1.711)

and d F (x; α, β, a), x > 0, dx

f (x; α, β, a) =

(1.1.712)

where α, β, a are all positive parameters. Remarks 1.258. (a) The formula Γ (a, z) = z∞ t a−1e−t dt will not result in F (x; α, β, a) to R be a cd f . The correct formula is Γ (a, z) = 0z t a−1 e−t dt. (b) Taking G (x) = 1 − (1 + xα )−β , x ≥ 0, the cd f (1.1.711) can be expressed as R

1 F (x; α, β, a) = Γ (a)

Z − log[1−G(x)] 0

t a−1e−t dt, x ≥ 0,

which is a special case of the cd f discussed in Remark 1.24.

1.1.355

Beta Kumarswamy Burr Type X (BKBX)

The cd f and pd f of BKBX are given, respectively, by 1 F (x; τ, υ, ϕ, ψ, ν, κ) = B (ν, κ)

Z 1− 1−e−(τx)2 0



υϕψ

t ν−1 (1 − t)κ−1 dt, x ≥ 0,

(1.1.713)

and d F (x; τ, υ, ϕ, ψ, ν, κ), x > 0, dx where τ, υ, ϕ, ψ, ν, κ are all positive parameters. f (x; τ, υ, ϕ, ψ, ν, κ) =

(1.1.714)

Remarks 1.259. (a) Why is it necessary to have ”υϕψ ” as the exponent?Why not just  θ= υϕψ and reduce the number of parameters to four. (b) Taking G (x) = 1− 1 − e−(τx) x ≥ 0, the cd f (1.1.713) can be expressed as F (x; τ, υ, ϕ, ψ, ν, κ) =

1 B (ν, κ)

Z G(x) 0

which has been discussed in Remark 1.15.

t ν−1 (1 − t)κ−1 dt, x ≥ 0, , x ≥ 0,

2

υϕψ

,

Introduction

1.1.356

167

Beta Kumaraswamy Marshall-Olkin-G (BKwMO-G)

The cd f and pd f of BKwMO-G are given, respectively, by

F (x; α, a, b, m, n) =

1 × B (m, n) h

Z 1− 1− 0



a ib G(x) 1−αG(x)

t m−1 (1 − t)n−1 dt, x ∈ R,

(1.1.715)

and d F (x; α, a, b, m, n), x ∈ R, (1.1.716) dx where α, a, b, m, n are all positive parameters, α = 1 − α and G (x) is a baseline cd f with the corresponding pd f g (x). f (x; α, a, b, m, n) =

Remark 1.260. This is the same distribution reported by the same authors in 1.1.351.

1.1.357

Poisson Burr X Burr XII (PBXBXII)

The cd f and pd f of PBXBXII are given, respectively, by

F (x; λ, θ, a, b) =

"

#   h i2 θ a b 1 − exp −λ 1 − exp − 1 − (1 + x ) 1 − e−λ

,

(1.1.717)

x ≥ 0 and d F (x; λ, θ, a, b), x > 0, (1.1.718) dx where λ, θ, a, b are all positive parameters.   h i2 θ a b Remark 1.261. Taking G (x) = 1 − exp − 1 − (1 + x ) , x ≥ 0, the cd f f (x; λ, θ, a, b) =

(1.1.717) can be expressed as

F (x; λ, θ, a, b) = which is discussed in Remark 1.40.

1.1.358

1 − exp [−λG (x)] , x ≥ 0, 1 − e−λ

Transmuted Arcsine (TA)

The cd f and pd f of TA are given, respectively, by     π + 2 arcsin(x) π − 2 arcsin(x) F (x; λ) = 1+λ , 2π 2π

(1.1.719)

G. G. Hamedani

168 −1 ≤ x ≤ 1 and f (x; λ) =

d F (x; λ) , − 1 < x < 1, dx

(1.1.720)

where λ > 0 is a parameter. Remark 1.262. Taking G (x) = written as

1 2

+ π1 arcsin(x) , −1 ≤ x ≤ 1, the cd f (1.1.719) can be

F (x; λ) = (1 + λ) G (x) − λG (x)2 , − 1 ≤ x ≤ 1,

which is taken up in Remark 1.28.

1.1.359

Poisson Burr X Generalized Lomax (PBXGL)

The cd f and pd f of PBXGL are given, respectively, by

F (x; a, δ, θ, γ) =

)   n o−2 δ  −γ −θ 1 − exp −a 1 − exp − 1 − (x + 1) −1 (

1 − e−a

, (1.1.721)

x ≥ 0, and d F (x; a, δ, θ, γ), x > 0, (1.1.722) dx where a, δ, θ, γ are all positive parameters.   n o−2 δ  −γ −θ Remark 1.263. Taking G (x) = 1 − exp − 1 − (x + 1) −1 , x ≥ 0, the f (x; a, δ, θ, γ) =

cd f (1.1.721) can be expressed as

F (x; a, δ, θ, γ) = (1 + a) G (x) − aG (x)2 , x ≥ 0, which is discussed in Remark 1.28.

1.1.360

Transmuted Power Function (TPF)

The cd f and pd f of TPF are given, respectively, by  α   α  x x F (x; α, β, θ) = 1+θ−θ , 0 ≤ x ≤ β, β β

(1.1.723)

and

d F (x; α, β, θ), 0 < x < β, dx where α, β, θ are all positive parameters. f (x; α, β, θ) =

(1.1.724)

Introduction Remark 1.264. Taking G (x) =

 α x β

169

, 0 ≤ x ≤ β, the cd f (1.1.723) can be expressed as

F (x; α, β, θ) = (1 + θ) G (x) − θG (x)2 , 0 ≤ x ≤ β, which is discussed in Remark 1.28.

1.1.361

Kumaraswamy Moment Exponential (KwME)

The cd f and pd f of KwME are given, respectively, by

x ≥ 0 and

 a b x −x/β F (x; β, a, b) = 1 − 1 − 1 − 1 + e , β 

f (x; β, a, b) =





d F (x; β, a, b), x > 0, dx

(1.1.725)

(1.1.726)

where β, a, b are all positive parameters. h   ia Remark 1.265. Taking G (x) = 1 − 1 + βx e−x/β , x ≥ 0, the cd f (1.1.725) can be written as F (x; β, a, b) = 1 − {1 − G (x)}b , x ≥ 0,

which is a special case of the cd f mentioned in Remark 1.41.

1.1.362

Marshall-Olkin Length Biased Exponential (MOLBE)

The cd f and pd f of MOLBE are given, respectively, by   1 − 1 + βx e−x/β   F (x; β, γ) = , x ≥ 0, 1 − (1 − γ) 1 + βx e−x/β

(1.1.727)

and

f (x; β, γ) =

d F (x; β, γ), x > 0, dx

(1.1.728)

where β, γ are positive parameters.   Remark 1.266. Taking G (x) = 1 − 1 + βx e−x/β , x ≥ 0, the cd f (1.1.727) can be expressed as F (x; β, γ) =

G (x) , x ≥ 0, 1 − (1 − γ) G (x)

which is a special case of the cd f mentioned in Remark 1.53.

G. G. Hamedani

170

1.1.363

Generalized Moment Exponential Power Series (GMEPS)

The cd f and pd f of GMEPS are given, respectively, by F (x; α, β, θ) = 1 −

M (θH (x)) , x ≥ 0, M (θ)

(1.1.729)

and f (x; α, β, θ) =

θg (x; α, β)M 0 (θH (x)) , x > 0, M (θ)

(1.1.730) α

z α −βx and where α, β, θ are all positive parameters, M (θ) = ∑∞ z=1 az θ , H (x) = (1 + βx )e α g (x; α, β) = αβ2 x2α−1 e−βx .

Remarks 1.267. The following important points were not mentioned in the paper: (a) According to equation (3) on page 2, M (θ) > 0, az ≥ 0 for all z ∈ R and θ is chosen so that M (θ) < ∞. (b) In equation (5) on page 3, M 0 (θH (x)) should be M (θH (x)) and it should be mention that M (θ) is increasing function of θ. (c) M 0 (θH (x)) = −θg (x; α, β) d on page 3, line 1_ should be dx M (θH (x)) = −θg (x; α, β)M 0 (θH (x)).

1.1.364

Truncated Exponential Skew Logistic (TESL)

The cd f and pd f of TESL are given, respectively, by h
−1 i 1 − exp −λ 1 + e−(x−µ)/σ , x ∈ R, F (x; λ, σ, µ) = 1 − e−λ and

(1.1.731)

d F (x; λ, σ, µ), x > 0, (1.1.732) dx where λ > 0, σ > 0, µ ∈ R are parameters.
−1 Remark 1.268. Taking G (x) = 1 + e−(x−µ)/σ , x ∈ R, the cd f (1.1.731) can be written as f (x; λ, σ, µ) =

1 − exp [−λG (x)] , x ≥ 0, 1 − e−λ which has been taken up in Remark 1.40. F (x; λ, σ, µ) =

1.1.365

Balakrishnan Alpha Skew Normal2 (BASN2)

The cd f and pd f of BASN2 are given, respectively, by F (x; α) = and

Z x

−∞

f (u; α)du, x ∈ R,

(1.1.733)

Introduction

171

" #2 (1 − αx)2 + 1 1 φ (x) , x ∈ R, f (x; α) = C2 (α) 2 + α2

(1.1.734)

where α ∈ R is a parameter, φ (x) is the pd f of the standard normal distribution and 4 C2 (α) = 3 − 2+α 2.

1.1.366

Alpha Beta Skew Logistic-G (ABSLG)

The cd f and pd f of ABSLG are given, respectively, by F (x; α, β) =

Z x

−∞

and f (x; α, β) =

h

f (u; α, β)du, x ∈ R,

1 − αx − βx3

i + 1 e−x

C2 (α, β)(1 + e−x )2

where α ∈ R, β ∈
R are 2 2 4 6 2 210 + 35π α + 98π αβ + 155π β /105.

1.1.367


2

(1.1.735)

, x ∈ R,

parameters

and

(1.1.736) C2 (α, β)

=

Bimodal Alpha Skew LogisticG2 (BASLG2)

The cd f and pd f of BASLG2 are given, respectively, by F (x; α) =

Z x

and f (x; α) =

−∞

f (u; α)du, x ∈ R,

h i (1 − αx)2 + 1 e−x C2 (α)(1 + e−x )2

, x ∈ R,

(1.1.737)

(1.1.738)

where α ∈ R is a parameters and C2 (α) is the normalizing constant.

Remark 1.269. The pd f (1.1.738) is a special case of the pd f (1.1.736).

1.1.368

Generalized Modified Exponential-G (GMEG)

The cd f and pd f of GMEG are given, respectively, by F (x; β, θ, η) =

Z

G(x;η)θ 1−G(x;η)θ

0

u −u/β e du, x ≥ 0, β2

(1.1.739)

and d F (x; β, θ, η), x > 0, (1.1.740) dx where β > 0, θ > 0 are parameters and G (x; η) is a baseline cd f which may depend on the parameter η. f (x; β, θ, η) =

G. G. Hamedani

172

Remark 1.270. Taking K (x) = G (x; η)θ , x ≥ 0, the cd f (1.1.739) can be rewritten as K(x) K(x)

Z

F (x; β, θ, η) =

0

u −u/β e du, x ≥ 0, β2

and with the change of variables t = βu ,we arrive at F (x; β, θ, η) =

Z

K(x) K(x)

0

te−t du, x ≥ 0,

which is a special case of the cd f discussed in Remark 1.37.

1.1.369

Doubly Truncated Extreme Value Type I (DTEVTI)

The cd f and pd f of DTEVTI (WLOFG for µ = 0 and θ = 1) are given, respectively, by −x

and

−A

e−e − e−e F (x) = −e−B A ≤ x ≤ B, −A , e − e−e −x

e−e e−x f (x) = −e−B A < x < B. −A , e − e−e

1.1.370

(1.1.741)

(1.1.742)

Lomax Exponential (LE)

The cd f and pd f of LE are given, respectively, by xex F (x; a, b) = 1 − 1 + b 

and



−a

, x ≥ 0,

  x −a−1 a (x + 1) ex xe f (x; a, b) = 1+ , x > 0, b b

(1.1.743)

(1.1.744)

where a > 0, b > 0 are parameters. Remark 1.271. Taking G (x) =

xex b+xex ,

x ≥ 0, the cd f (1.1.743) can be expressed as

G (x) F (x; a, b) = 1 − 1 + 1 − G (x) 

which is a special case of the cd f (1.1.57).



−a

, x ≥ 0,

Introduction

1.1.371

173

Generalized Odd Log-Logistic Exponential (GOLLEx)

The cd f and pd f of GOLLEx are given, respectively, by αθ 1 − e−λx F (x; α, θ, λ) =  αθ n  θ oα , x ≥ 0, 1 − e−λx + 1 − 1 − e−λx 

and

f (x; α, θ, λ) =

d F (x; α, θ, λ), x > 0, dx

(1.1.745)

(1.1.746)

where α, θ, λ are all positive parameters. Remark 1.272. The cd f (1.1.323) with an additional parameter λ is  α −β   b a   1 − e−λx  
, x ≥ 0. F (x; α, β, λ, a, b) = 1 − 1 + b a   1 − 1 − e−λx     

Taking β = 1, F (x; α, 1, λ, a, b) can be rewritten as h i b αa 1 − e−λx F (x; α, 1, λ, a, b) =      , x ≥ 0, b αa b a α 1 − e−λx + 1 − 1 − e−λx

which for b = 1, reduces to the cd f (1.1.745).

1.1.372

Extended Odd Weibull Exponential (EOWEx)

The cd f and pd f of EOWEx are given, respectively, by

and

n  α o−1/β F (x; α, β, λ) = 1 − 1 + eλx − 1 , x ≥ 0, f (x; α, β, λ) =

d F (x; α, β, λ), x > 0, dx

(1.1.747)

(1.1.748)

where α, β, λ are all positive parameters. Remark 1.273. Taking G (x) = 1 − e−λx , x ≥ 0, the cd f (1.1.747) can be written as G (x) F (x; α, β, λ) = 1 − 1 + β 1 − G (x) 



which is the same as the cd f given in Remark 1.20.

α −1/β

, x ≥ 0,

174

1.1.373

G. G. Hamedani

New Libby-Novick (NLN)

The cd f and pd f of NLN are given, respectively, by F (x; α, β, c) = 1 − and

(1 − xα )β

(1 − (1 − c) xα )β

, 0 ≤ x ≤ 1,

d F (x; α, β, c), 0 < x < 1, dx where α, β, c are all positive parameters. f (x; α, β, c) =

(1.1.749)

(1.1.750)

Remark 1.274. Taking G (x) = xα , 0 < x < 1, the cd f (1.1.749) can be written as F (x; α, β, λ) = 1 −

G(x)β (1 − (1 − c) G (x))β

, 0 ≤ x ≤ 1,

which is the same as the cd f given in Remark 1.20.

1.1.374

Modified Burr XII (MBXII)

The cd f and pd f of MBXII are given, respectively, by

and

 −k , x ≥ 0, F (x; λ, c, k) = 1 − 1 + xc eλx f (x; λ, c, k) =

d F (x; λ, c, k), x > 0, dx

(1.1.751)

(1.1.752)

where λ, c, k are all positive parameters. Remark 1.275. Taking G (x) = 1 + x−c/α eλx/α one given in Remark 1.205.

1.1.375


−1

, the cd f (1.1.751) will be similar to the

Poisson Odd Generalized Exponential (POGE)

The cd f and pd f of POGE are given, respectively, by −λ 1−e

F (x; α, β, λ, ζ) =

1−e

and

−α

1 − e−λ

G(x) G(x)

!β

, x ≥ 0,

d F (x; α, β, λ, ζ), x > 0, dx where α, β, λ, ζ are all positive parameters and G (x) is a baseline cd f . f (x; α, β, λ, ζ) =

(1.1.753)

(1.1.754)

Introduction

175

 β G(x) −α G(x) Remark 1.276. Taking K (x) = 1 − e , x ≥ 0, the cd f (1.1.753) will be written as F (x; α, β, λ, ζ) = which has been discussed in Remark 1.62.

1.1.376

1 − e−λK(x) , x ≥ 0, 1 − e−λ

Modified Odd Weibull-G (MOW-G)

The cd f and pd f of MOW-G are given, respectively, by n oθ G(x;ζ) −λ 1−G(x;ζ)[1+G(x;ζ)]/2

F (x; λ, θ, ζ) = 1 − e

, x ∈ R,

(1.1.755)

and

f (x; λ, θ, ζ) =

h i λθg (x; ζ) G (x; ζ)θ−1 1 + G (x; ζ)2 /2 {1 − G (x; ζ) [1 + G (x; ζ)] /2}θ+1

oθ n G(x;ζ) −λ 1−G(x;ζ)[1+G(x;ζ)]/2

e

× (1.1.756)

,

x ∈ R, where λ, θ, ζ are all positive parameters and G (x; ζ) is a baseline cd f with the corresponding pd f g (x; ζ).

1.1.377

Generalized Uniform (GU)

The cd f and pd f of GU are given, respectively, by F (x; α, θ) = and f (x; α, θ) =

1 − αθ [x (1 − α) + α]−θ , 0 ≤ x ≤ 1, 1 − αθ (1 − α) θαθ

(1 − αθ ) [x (1 − α) + α]θ+1

, 0 < x < 1,

(1.1.757)

(1.1.758)

where α > 0, θ > 0 are positive parameters.

1.1.378

McDonald Modified Burr-III (McMB-III)

The cd f and pd f of McMB-III are given, respectively, by F (x; α, β, γ, λ, η, ζ) = x ≥ 0, and

Z x 0

f (x; α, β, γ, λ, η, ζ)du

(1.1.759)

G. G. Hamedani

176

λζα

− γ −1 ζαβx−β−1  1 + γx−β × B (λ, η) !  − ζαγ −1 η−1 −β 1 − 1 + γx ,

f (x; α, β, γ, λ, η, ζ) =

(1.1.760)

x > 0, where α, β, γ, λ, η, ζ are all positive parameters.

1.1.379

Generalized Lindley (GL)

The cd f and pd f of GL are given, respectively, by α (1 + a) eax − F (x; α, β, a) = 1 − α (1 + a + ax) α 

and

f (x; α, β, η) = 

βa2 (1 + a) (1 + x) eax (1 + a + ax)2

(1 + a) eax α − α (1 + a + ax) α

−β

, x ≥ 0,

(1.1.761)

×

−β−1

, x > 0,

(1.1.762)

where α, β, a are all positive parameters.

1.1.380

Weibull Marshall-Olkin Lindley (WMOL)

The cd f and pd f of WMOL are given, respectively, by (   β ) (1 + a) eax α F (x; α, β, a) = 1 − exp − log − , α (1 + a + ax) α

(1.1.763)

x ≥ 0 and

f (x; α, β, a) =

h  iβ−1 (1+a)eax βa2 (1 + a) (1 + x) eax log α(1+a+ax) − αα

(1 + a + ax) [(1 + a) eax − α (1 + a + ax)] (   β ) (1 + a) eax α × exp − log − , x > 0, α (1 + a + ax) α

where α, β, a are all positive parameters.

(1.1.764)

Introduction

1.1.381

177

Generalized Inverse Marshall-Olkin (GIMO)

The cd f and pd f of GIMO are given, respectively, by

and

h iθ (1 + c)θ − 1 + cG (1/x)β F (x; β, θ, c) = h ih iθ , x ≥ 0, θ β (1 + c) − 1 1 + cG (1/x)

f (x; β, θ, c) = h

βθc (1 + c)θ x−2 g (1/x)G (1/x)β−1  , x > 0, i  θ β β 1 + cG (1/x) 1 + cG (1/x) −1

(1.1.765)

(1.1.766)

where β, θ, c are all positive parameters and G (x) is a baseline cd f with the corresponding pd f g (x). Remark 1.277. The authors considered the following F (x; β, 1, c); ( h sub-models: iθ ) −(λ/x)β

1− (1−α)e αθ ; F x; 1, 1, 1−α ; F (x; α, θ, λ, β) = 1−α h i . Clearly simiF x; 1, θ, 1−α θ α α β θ (1−α)e−(λ/x)

lar characterizations can be stated for the sub-models.

1.1.382

Poisson Burr X Inverse Rayleigh (PBX-IR)

The cd f and pd f of PBX-IR are given, respectively, by     2 θ 2 x−2   −α   − e −α2 x−2 1−e   1 − exp −λ 1 − e     F (x; α, θ, λ) = , x ≥ 0, 1 − e−λ and d F (x; α, θ, λ), x > 0, dx where α, θ, λ are all positive parameters. f (x; α, θ, λ) =

−

Remark 1.278. Taking G (x) = 1 − e pressed as F (x; α, θ, λ) =



2 −2 e−α x 2 −2 1−e−α x

2

(1.1.768)

, x ≥ 0, the cd f (1.1.767) can be ex-

n o 1 − exp −λG (x)θ

which has been considered in Remark 1.40.

(1.1.767)

1 − e−λ

, x ≥ 0,

G. G. Hamedani

178

1.1.383

Kumarswamy Reciprocal (KR)

The cd f and pd f of KR are given, respectively, by "

b F (x; α, β, a, b) = 1 − 1 − ln a 

and

#β −α  x α ln , a ≤ x ≤ b, a

 −α  x α−1 b f (x; α, β, a, b) = αβ ln x−1 ln a a " #β−1   b −α  x α × 1 − ln ln , a < x < b, a a

(1.1.769)

(1.1.770)

where a > 0, b, α > 0 and β > 0 are parameters.

1.1.384

Log-Weighted Pareto (LWP)

The cd f and pd f of LWP are given, respectively, by   α  1 + α logx β , x ≥ β > 1, F (x; α, β) = 1 − x 1 + α log β and α2 β2 logx , x > β, xα+1 (1 + α log β) where α > 0 and β > 1 are parameters. f (x; α, β) =

1.1.385

(1.1.771)

(1.1.772)

Alpha Power Transformed Extended Exponential (APTEE)

The cd f and pd f of APTEE are given, respectively, by   α1−e1−(1+λx)β −1 , α > 0, α 6= 1 α−1 , x ≥ 0, F (x; α, β, λ) = 1 − e1−(1+λx)β , α = 1

(1.1.773)

and

d F (x; α, β, λ), x > 0, dx where α, β, λ are all positive parameters. f (x; α, β, λ) =

1−(1+λx)β

Remark 1.279. Taking G (x) = 1 − e expressed as

F (x; α, β, λ) = which has been discussed in Remark 1.43.

(1.1.774)

, x ≥ 0, the cd f (1.1.773), for α 6= 1, can be

αG(x) − 1 , x ≥ 0, α−1

Introduction

1.1.386

179

Lomax Exponentiated Weibull (LEW)

The cd f and pd f of LEW are given, respectively, by   −θ β a 1 − e−x F (x; a, β, θ) = 1 − 1 +
 , x ≥ 0, β a 1 − 1 − e−x 

and

f (x; a, β, θ) =

d F (x; a, β, θ), x > 0, dx

(1.1.775)

(1.1.776)

where a, β, θ are all positive parameters. Remarks 1.280. (a)The cd f (1.1.775) can be expressed as h   i β a θ F (x; a, β, θ) = 1 − 1 − 1 − e−x , x ≥ 0,

which is a special case of the cd f discussed in Remark 1.4. (b) There is a typo in equation (1.1.1) of the paper; "x ∈ R" should be "x ≥ 0". (c) The second author is also a coauthor of the distribution mentioned in Remark 1.4.

1.1.387

Intervened Geometric Compound (IGC)

The cd f and pd f of IGC are given, respectively, by   G (x) 1 − ρθ2 G (x)

, x ∈ R, F (x; ρ, θ) = 1 − ρθG (x) 1 − θG (x)

(1.1.777)

and

h i 2 (1 − θ) (1 − ρθ) g (x) 1 − ρθ2 G (x)

, x ∈ R, f (x; ρ, θ) = [ 1 − ρθG (x) 1 − θG (x) ]2

where 0 < θ < 1, 0 ≤ ρ < corresponding pd f g (x).

1.1.388

1 θ

(1.1.778)

< ∞ are parameters and G (x) is a baseline cd f with the

Intervened Negative Binomial Compound (INBC)

The cd f and pd f of INBC are given, respectively, by (1 − θ)r (1 − ρθ)r F (x; ρ, θ, r) = 1 − 1 − (1 − θ)r 

x ∈ R, and

"

#  1 − (1 − θG (x))r

, 1 − ρθG (x) 1 − θG (x) 

(1.1.779)

G. G. Hamedani

180

 (1 − θ)r (1 − ρθ)r × f (x; ρ, θ, r) = 1 − (1 − θ)r h 
i  rθg (x) 1 + ρ − 2ρθG (x) − ρ 1 − θG (x) r+1  , 

r+1   1 − ρθG (x) 1 − θG (x) 

x ∈ R, where 0 < θ < 1, 0 ≤ ρ < with the corresponding pd f g (x).

1.1.389

1 θ

(1.1.780)

< ∞, r > 0 are parameters and G (x) is a baseline cd f

Intervened Binomial Compound (IBC)

The cd f and pd f of IBC are given, respectively, by  m 
m  ρθGx + 1 θG(x) + 1 − 1 F (x; ρ, θ, m) = 1 − , (ρθ + 1)m [(θ + 1)m − 1]

(1.1.781)

x ∈ R, and

 m−1 mθg (x) ρθGx + 1 f (x; ρ, θ, m) = × (ρθ + 1)m [(θ + 1)m − 1] n o
m−1   θGx + 1 2ρθG(x) + ρ + 1 − ρ ,

(1.1.782)

x ∈ R, where 0 < θ < 1, 0 ≤ ρ < 1θ < ∞, m ∈ N are parameters and G (x) is a baseline cd f with the corresponding pd f g (x).

1.1.390

Intervened Poisson Compound (IPC)

The cd f and pd f of IPC are given, respectively, by   eλρG(x) eλG(x) − 1
F (x; ρ, λ) = 1 − , eλρ eλ − 1

x ∈ R,

(1.1.783)

h i f (x; ρ, λ) = λg (x) eλρG(x) (1 + ρ) eλG(x) − ρ , x ∈ R,

(1.1.784)

and

where λ > 0, ρ ≥ 0 are parameters and G (x) is a baseline cd f with the corresponding pd f g (x). Remarks 1.281. Characterizations stated in the following sections for IPC can be stated for the following distributions proposed in Jayakumar and Sankaran (2020): (a) Exponentiated Weibull Intervened Poisson

F (x) = 1 −

−λ(1+ρ)

e



β 1−e−(θx)

α

    β α −λ 1+ρ 1−e−(θx)

−e 1 − e−λ

, x ≥ 0.

Introduction

181

(b) Fréchet Intervened Poisson F (x) = 1 −

−λ(1+ρ)e−(θx)

(c) Lomax Intervened Poisson

1.1.391

  β −λ 1+ρe−(θx)

−e 1 − e−λ

e

F (x) = 1 −

β

, x ≥ 0.

eλ(1+ρ)[1+(x/θ)] − eλρ[1+(x/θ)]
, x ≥ 0. eλρ eλ − 1

Odd Log-Logistic Exponential Gaussian (OLLExGa)

The cd f and pd f of OLLExGa are given, respectively, by F (x; µ, σ, ν, τ) =

G (x; µ, σ, ν)τ , G (x; µ, σ, ν)τ + G (x; µ, σ, ν)τ

(1.1.785)

x ∈ R, and d F (x; µ, σ, ν, τ), x ∈ R, (1.1.786) dx where µ ∈ R, G (x; µ, σ, ν) =  σ > 0, ν
> 0, τ > 0 are parameters, µ−1 t−µ 1Rx σ2 σ ν 0 exp ν + 2ν2 Φ σ − ν dt, x ∈ R and Φ (·) is the standard normal cumulative function. f (x; µ, σ, ν, τ) =

Remark 1.282. The characterizations of the cd f (1.1.785) are similar to those of the cd f (1.1.107).

1.1.392

Gompertz Fréchet (GFr)

The cd f and pd f of GF are given, respectively, by   n o−γ  θ −(α/x) F (x; α, β, γ, θ) = 1 − exp 1− 1−e , γ

(1.1.787)

x ≥ 0, and

d F (x; α, β, γ, θ), x > 0, dx where α, β, γ, θ are all positive parameters. f (x; α, β, γ, θ) =

(1.1.788)

Remark 1.283. Taking G (x) = e−(α/x), x ≥ 0, the cd f (1.1.787) can be expressed as    θ G (x)γ − 1 F (x; α, β, γ, θ) = 1 − exp , x ≥ 0, γ G (x)γ

which is similar to the one proposed by Yousof et al. (2016) and has been characterized in Hamedani (2019).

G. G. Hamedani

182

1.1.393

Weighted Exponential Gompertz (WE-G)

The cd f and pd f of WE-G are given, respectively, by F (x; α, β, σ, λ) = x ≥ 0, and

iβ  h iαβ  λx λx 1h 1 − e−σ(e −1) (α + 1) − 1 − e−σ(e −1) , α

(1.1.789)

d F (x; α, β, σ, λ), x > 0, (1.1.790) dx where α, β, σ, λ are all positive parameters. h iβ λx Remark 1.284. Taking G (x) = 1 − e−σ(e −1) , x ≥ 0, the cd f (1.1.789) can be expressed as    1 1 α F (x; α, β, σ, λ) = G (x) 1 + − G (x) , x ≥ 0, α α which is mentioned in Remark 1.110. f (x; α, β, σ, λ) =

1.1.394

Rayleigh Rayleigh (RR)

The cd f and pd f of RR are given, respectively, by −

and

f (x; β, σ) =

x4 8β4 σ2

x ≥ 0,

(1.1.791)

d F (x; β, σ) , x > 0, dx

(1.1.792)

F (x; β, σ) = 1 − e

,

where β, σ are positive parameters. Remark 1.285. The cd f (1.1.791) can be rewritten as 4

F (x; β, σ) = 1 − e−λ(x/β) , x ≥ 0,

which is simply a Weibull distribution.

1.1.395

Chen-G (CG)

The cd f and pd f of CG are given, respectively, by F (x; β, λ) = and

  β λ 1−eG(x)

1−e 1 − eλ(1−e)

,

x ∈ R,

d F (x; β, λ) , x ∈ R, dx where β, λ are positive parameters and G (x) is a baseline cd f . f (x; β, λ) =

(1.1.793)

(1.1.794)

Introduction Remark 1.286. Taking K (x) = expressed as

1−eG(x) 1−e

β

183

, x ∈ R and γ = λ (e − 1), the cd f (1.1.793) can be

1 − e−γK(x) , x ∈ R, 1 − e−γ which is has been discussed in Remark 1.113. F (x; β, λ) =

1.1.396

Power Muth (PM)

The cd f and pd f of PM are given, respectively, by n  o γ F (x; β, γ) = 1 − exp (x/β)γ − e(x/β) − 1 , x ≥ 0,

(1.1.795)

and

f (x; β, γ) =

  γ γxγ−1 e(x/β) − 1 βγ

n  o γ exp (x/β)γ − e(x/β) − 1 , x > 0,

(1.1.796)

where β, γ are positive parameters.

1.1.397

Cubic Transmuted Power Function (CTPF)

The cd f and pd f of CTPF are given, respectively, by F (x; α, β, λ) =

1 + λ α 2λ 2α λ x − 2α x + 3α x3α , α β β β

0 ≤ x ≤ β,

(1.1.797)

and d F (x; α, β, λ), 0 < x < β, (1.1.798) dx where α > 0, β > 0 and λ ∈ [−1, 1] are parameters.  α Remark 1.287. Taking G (x) = βx , 0 ≤ x ≤ β, the cd f (1.1.797) can be written as f (x; α, β, λ) =

F (x; α, β, λ) = (1 + λ) G (x) − 2λG (x)2 + λG (x)3 ,

which has been characterized in Hamedani (2019).

1.1.398

Weibull Alpha Power Inverted Exponential (WAPIE)

The cd f and pd f of WAPIE are given, respectively, by  " #β  −λ/x  αe − 1  , x ≥ 0, F (x; α, β, λ) = 1 − exp −ϕ −λ/x   α − αe

and

(1.1.799)

G. G. Hamedani

184

f (x; α, β, λ) =



−λ/x

αe

−1 α−1

β−1

ϕβλ lnα −λ/x e−λ/x e α h i × −λ/x β+1 x2 (α − 1) α−αe α−1

 " #β  −λ/x  αe − 1  exp −ϕ , −λ/x   α − αe

(1.1.800)

x > 0, where α > 0, α 6= 1, β > 0 and λ > 0 are parameters.

1.1.399

Exponentiated-Epsilon (E-Epsilon)

The cd f and pd f of E-Epsilon are given, respectively, by "  −λd/2#α d +x F (x; α, λ, d) = 1 − , 0 ≤ x ≤ d, d −x

(1.1.801)

and   αλd 2 d + x −λd/2 f (x; α, λ, d) = 2 × d − x2 d − x #α−1 "   d + x −λd/2 1− , 0 < x < d, d −x

(1.1.802)

where α > 0, λ > 0 and d > 0 are parameters.

1.1.400

Gompertz-Alpha Power Inverted Exponential (GAPIE)

The cd f and pd f of GAPIE are given, respectively, by ( "  b #) a α−1 F (x; α, a, b, c) = 1 − exp 1− , x ≥ 0, −c/x b α − αe

(1.1.803)

and ac logα −c/x e−c/x f (x; α, a, b, c) = e α (α − 1) x2



α−1 −c/x α − αe

b−1

where α > 0 (α 6= 1), a > 0, b > 0 and d > 0 are parameters.

,

x > 0,

(1.1.804)

Introduction

1.1.401

185

Gompertz Extended Generalized Exponential (G-EGE)

The cd f and pd f of G-EGE are given, respectively, by ( "  γ λ #) β − (β − 1)γ θ 1− F (x; β, γ, λ, θ) = 1 − exp , λ βγ − (β − e−x )γ

(1.1.805)

x ≥ 0, and
λ
γ−1 f (x; β, γ, λ, θ) = θγe−x βγ − (β − 1)γ β − e−x × h i
γ −(λ+1) βγ − β − e−x × ( " λ #)  γ θ β − (β − 1)γ exp 1− , x > 0, λ βγ − (β − e−x )γ

(1.1.806)

where β > 1, γ > 1, λ > 0 and θ > 0 are parameters.

1.1.402

Extended Odd Log-Logistic (EOLL-G)

The cd f and pd f of EOLL-G are given, respectively, by (   F (x; α, β, η) = exp − − log

G (x; η)α G (x; η)α + G (x; η)α

β )

, x ∈ R,

(1.1.807)

and αβg (x; η) G (x; η)α−1  × G (x; η) G (x; η)α + G (x; η)α   β−1 G (x; η)α − log × G (x; η)α + G (x; η)α (   β ) G (x; η)α exp − − log , G (x; η)α + G (x; η)α

f (x; α, β, η) =

(1.1.808)

x ∈ R, where α > 0, β > 0 are parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η), which may depend on a parameter vector η.

1.1.403

Generalized Odd Generalized Exponential G (GOGE-G)

The cd f and pd f of GOGE-G are given, respectively, by F (x; α, β, λ) =

1 − exp

(

−λ

−e−αx −λ 1 − e−αx

)!β

, x ≥ 0,

(1.1.809)

G. G. Hamedani

186 and

d F (x; α, β, λ), x > 0, dx where α, β, λ are all positive parameters. f (x; α, β, λ) =

(1.1.810)

Remark 1.288. The cd f (1.1.809) is the same as the cd f (1.1.559).

1.1.404

Slash Maxwell (SM)

The cd f and pd f of SM are given, respectively, by F (x; σ, p) =

Z x 0

f (x; σ, p) , x ≥ 0,

(1.1.811)

and

f (x; σ, p) =

2pΓ



p+3 2



  x −(p+1)  p + 3 2 2 G x ; , σ , x > 0, Γ (1/2) σ 2

where σ, p are all positive parameters and G (x; α, β) =

1.1.405

(1.1.812)

R x α−1 −t/β 1 e dt. βα Γ(α) 0 t

Modified T-X (MT-X)

The cd f and pd f of MT-X are given, respectively, by F (x; λ, η) = (1 − λ) G (x; η) + 3λG (x; η)2 − 2λG (x; η)3 , x ∈ R,

(1.1.813)

and d F (x; η) , x ∈ R, (1.1.814) dx where |λ| ≤ 1 is a parameter and G (x; η) is a baseline cd f which may depend on the parameter vector η. f (x; λ, η) =

Remark 1.289. The cd f (1.1.813) is the same as the cd f (1.1.465).

1.1.406

Double Truncated Transmuted Fréchet (DTTF)

The cd f and pd f of DTTF are given, respectively, by h i −β −β (1 + λ) e−αx − λe−αm

F (x; α, β, λ, m, g) = − [(1 + λ) A − λA2 ] − [(1 + λ) B − λB2 ] h i −β −β (1 + λ) e−αm − λe−2αm ,

(1.1.815)

Introduction

187

m ≤ x ≤ g, and

f (x; α, β, λ, m, g) =

αβx−β−1 e−αx

−β

h i −β (1 + λ) − 2λe−αx

[(1 + λ) A − λA2 ] − [(1 + λ) B − λB2 ]

, m < x < g,

(1.1.816)

where α > 0, β > 0, |λ| ≤ 1 are parameters and m and g are lower and upper truncation points.

1.1.407

Burr-Hatke Logarithmic BurrXII (BH-BXII)

The cd f and pd f of BH-BXII are given, respectively, by   1 − 1 − (1 + xα1 )−α2    , x ≥ 0, F (x; α1 , α2 ) = 1 − 1 − log 1 − 1 − (1 + xα1 )−α2

(1.1.817)

and

     1 − log 1 − 1 − (1 + xα1 )−α2 +1 f (x; α1 , α2 ) =   
2 , α2 +1 −α α 2 1 (1 + x ) 1 − log 1 − 1 − (1 + xα1 ) α1 α2 xα1 −1

(1.1.818)

x > 0, where α1 > 0, α2 > 0 are parameters.

1.1.408

Odd Inverse Pareto-Exponential (OIPEx)

The cd f and pd f of OIPEx are given, respectively, by

and

F (x; α, β, λ) = 



1 − e−λx

α

1 − (1 − β) e−λx

α , x ≥ 0,

d F (x; α, β, λ), x > 0, dx where α, β, λ are all positive parameters. f (x; α, β, λ) =

(1.1.819)

(1.1.820)

Remark 1.290. Taking G (x) = 1 − e−λx , x ≥ 0, the cd f (1.1.819) can be written as  α [G (x)]α G (x) F (x; α, β, λ) =  , x ≥ 0, α = β + (1 − β) G (x) 1 − (1 − β) G (x)

which is a special case of the cd f (1.1.23), as well as the cd f mentioned in Remark 1.8.

G. G. Hamedani

188

1.1.409

Poisson Rayleigh Generalized Lomax (PRGLx)

The cd f and pd f of PRGLx are given, respectively, by 1 × F (x; a, b, c, λ) = −λ   1 − e  !−2     h  x i−a −c 1 − exp −λ 1 − exp −   , 1− 1+ −1   b

(1.1.821)

x ≥ 0, and

d F (x; a, b, c, λ), x > 0, (1.1.822) dx where a, b, c, λ are all positive parameters. "  −2 # n o−c
  −a −1 , x ≥ 0, the Remark 1.291. Taking G (x) = 1 − exp − 1 − 1 + bx f (x; a, b, c, λ) =

cd f (1.1.821) can be expressed as

F (x; a, b, c, λ) = which has been discussed in Remark 1.40.

1.1.410

1 − e−λG(x) , x ≥ 0, 1 − e−λ

Generalized Odd Log-Logistic-G (GOLL-G)

The cd f and pd f of GOLL-G are given, respectively, by F (x; a1 , a2 , ψ) = and

G (x; ψ)a1 a2 , x ∈ R, G (x; ψ) + [1 − G (x; ψ)a2 ]a1 a1 a2

(1.1.823)

d F (x; a1 , a2 , ψ), x ∈ R, (1.1.824) dx where a1 , a2 are positive parameters and G (x; ψ) is a baseline cd f which may depend on the parameter vector ψ. f (x; a1 , a2 , ψ) =

Remark 1.292. Taking K (x) = G (x; ψ)a2 , x ∈ R, the cd f (1.1.823) can be written as F (x; a1 , a2 , ψ) =

K (x)a1 , x ∈ R, K (x)a1 + K (x)a1

which has been discussed in Remark 1.39.

Introduction

1.1.411

189

Marshall-Olkin Generalized Pareto (MOGP)

The cd f and pd f of MOGP are given, respectively, by F (x; α, θ, λ) = 1 − and

α 1 + θx λ


−1/θ

1 − α 1 + θx λ


−1/θ , x ≥ 0,

(1.1.825)

d F (x; α, θ, λ), x > 0, (1.1.826) dx where α, θ, λ are positive parameters and α = 1 − α.
−1/θ , x ≥ 0, the cd f (1.1.825) can be expressed Remark 1.293. Taking G (x) = 1 − 1 + θx λ as f (x; α, θ, λ) =

F (x; α, θ, λ) =

G (x) , x ≥ 0, G (x) + αG (x)

which has been discussed in Remark 1.39.

1.1.412

Log-Balakrishnan-Alpha-Skew-Normal (LBASN2 (α))

The cd f and pd f of LBASN2 (α) are given, respectively, by F (x; α) =

Z x 0

f (u; α)du, x ≥ 0,

(1.1.827)

and

f (x; α) =

h i2 (1 − α logx)2 + 1 φ (logx) xC2 (α)

, x > 0,

(1.1.828)

where α ∈ R is a parameter, C2 (α) = 4 + 8α2 + 3α4 and φ (x) is the pd f of the standard normal distribution Φ (x).

1.1.413

Flexible Additive Weibull (FAW)

The cd f and pd f of FAW are given, respectively, by F (x; a, b, c, d, e, g) = 1 − e−(ax

b +cxd +exg

) , x ≥ 0,

(1.1.829)

and   b d g f (x; a, b, c, d, e, g) = abxb−1 + cdxd−1 + egxg−1 e−(ax +cx +ex ), x > 0,

where a, b, c, d, e, g are all positive parameters.

(1.1.830)

190

1.1.414

G. G. Hamedani

Transmuted Weibull (TW)

The cd f and pd f of TW are given, respectively, by h ih i a a F (x; a, b, λ) = 1 − e−(x/b) 1 + λe−(x/b) , x ≥ 0,

(1.1.831)

and

d F (x; a, b, λ), x > 0, dx where a > 0, b > 0 and λ ∈ [−1, 1] are parameters. f (x; a, b, λ) =

(1.1.832)

a

Remark 1.294. Taking G (x) = 1 − e−(x/b) , x ≥ 0, the cd f (1.1.831) can be expressed as F (x; a, b, λ) = (1 + λ) G (x) − λG (x)2 , x ≥ 0,

which has been discussed in Remark 1.177.

1.1.415

Cubic Transmuted (CT)

The cd f and pd f of CT are given, respectively, by

x ∈ R, and

F (x; λ1 , λ2 ) = (1 + λ1 ) G (x) + (λ2 − λ2 ) G (x)2 − λ2 G (x)3 ,

(1.1.833)

d F (x; λ1 , λ2 ) , x ∈ R, (1.1.834) dx where λ1 ∈ [−1, 1], λ2 ∈ [−1, 1] and −2 ≤ λ1 +λ2 ≤ 1 are parameters and G (x) is a baseline cd f . f (x; λ1 , λ2 ) =

Remark 1.295. The cd f (1.1.833) has been characterized in Hamedani (2019).

1.1.416

Transmuted Modified Weibull (TMW)

The cd f and pd f of TMW are given, respectively, by h ih i β β F (x; α, β, η, λ) = 1 − e−αx−ηx 1 + λe−αx−ηx ,

(1.1.835)

x ≥ 0, and

d F (x; α, β, η, λ), x > 0, dx where α > 0, β > 0, η > 0 and λ ∈ [−1, 1] are parameters. f (x; α, β, η, λ) =

Remark 1.296. The cd f (1.1.835) is a special case of the cd f (1.1.689).

(1.1.836)

Introduction

1.1.417

191

Transmuted Half Normal (THN)

The cd f and pd f of THN are given, respectively, by      x x √ √ F (x; σ, λ) = erf 1 + λ 1 − erf , σ 2 σ 2 x ∈ R, and

(1.1.837)

d F (x; σ, λ) , x ∈ R, (1.1.838) dx where σ > 0 and λ ∈ [−1, 1] are parameters and erf (·) is the error function.   Remark 1.297. Taking G (x) = erf σ√x 2 , x ∈ R, the cd f (1.1.837) can be written as f (x; σ, λ) =

F (x; a, b, λ) = (1 + λ) G (x) − λG (x)2 , x ∈ R,

which was discussed in Remark 1.177.

1.1.418

Cubic Rank Transmuted Fréchet (CRTF)

The cd f and pd f of CRTF are given, respectively, by

x ≥ 0, and

h i −α −α 2 F (x; α, σ, λ1 , λ2 ) = λ1 e−(x/σ) + (λ2 − λ1 ) e−(x/σ) h i −α 3 + (1 − λ2 ) e−(x/σ) ,

d F (x; σ, λ) , x > 0, dx where α > 0, σ > 0, λ1 ∈ [0, 1] and λ2 ∈ [−1, 1] are parameters. f (x; α, σ, λ1 , λ2 ) =

(1.1.839)

(1.1.840)

−α

Remark 1.298. Taking G (x) = e−(x/σ) , x ≥ 0, the cd f (1.1.839) can be expressed as F (x; α, σ, λ1 , λ2 ) = λ1 G (x) + (λ2 − λ1 )G (x)2 + (1 − λ2 ) G (x)3 , x ≥ 0,

which has been characterized in Hamedani (2019).

1.1.419

Weighted Garima (WG)

The cd f and pd f of WG are given, respectively, by F (x; θ, β) = 1 −

(θx)β e−θx + (θ + β + 1) Γ (β, θx) , x ≥ 0, (θ + β + 1) Γ (β)

(1.1.841)

and θβ xβ−1 (1 + θ + θx) e−θx , x > 0, (θ + β + 1) Γ (β) where θ, β are positive parameters. f (x; θ, β) =

(1.1.842)

G. G. Hamedani

192

1.1.420

Exponentiated Exponential Lomax (EEL)

The cd f and pd f of EEL are given, respectively, by 

−α θ  β −λ x+β



F (x; α, β, λ, θ) = 1 − 1 − 1 − e and

, x ≥ 0,

d F (x; α, β, λ, θ), x > 0, dx where α, β, λ, θ are all positive parameters. f (x; α, β, λ, θ) =

(1.1.843)

(1.1.844)

Remark 1.299. The cd f (1.1.843) can be written as 



 α θ −λ 1+ βx

F (x; α, β, λ, θ) = 1 − 1 − 1 − e

, x ≥ 0,

which is a special case of the cd f given in Remark 1.42.

1.1.421

Transmuted Half Normal (THN)

The cd f and pd f of THN are given, respectively, by      2 x x √ √ F (x; σ, λ) = (1 + λ) erf − λ erf , x ∈ R, σ 2 σ 2

(1.1.845)

and d F (x; σ, λ) , x ∈ R, dx where σ > 0 and λ ∈ [−1, 1] are parameters. f (x; σ, λ) =

(1.1.846)

Remark 1.300. The cd f (1.1.845) is the same as the cd f (1.1.837).

1.1.422

Transmuted Ishita (TI)

The cd f and pd f of TI are given, respectively, by     ηx (ηx + 2) −λx F (x; η, λ) = (1 + λ) 1 − 1 + e η3 + 2     ηx (ηx + 2) −λx 2 −λ 1− 1+ e , η3 + 2

(1.1.847)

x ≥ 0, and d F (x; η, λ) , x > 0, dx where η > 0 and λ ∈ [−1, 1] are parameters. f (x; η, λ) =

(1.1.848)

Introduction 193     ηx(ηx+2) Remark 1.301. Taking G (x) = 1 − 1 + η3 +2 e−λx , x ≥ 0 , the cd f (1.1.847) can be written as F (x; η, λ) = (1 + λ) G (x) − λG (x)2 , x ≥ 0, which was discussed in Remark 1.28.

1.1.423

Transmuted Generalized Extreme Value (TGEV)

The cd f and pd f of TGEV are given, respectively, by

and

 2 x−µ x−µ F (x; µ, σ, γ, λ) = (1 + λ) e{−(1+γ( σ ))} − λ e{−(1+γ( σ ))} , f (x; µ, σ, γ, λ) =

for x such that 1 + γ eters.

x−µ
σ

d F (x; µ, σ, γ, λ), dx

(1.1.849)

(1.1.850)

> 0 , where µ ∈ R, σ > 0, γ > 0 (γ 6= 0), λ ∈ [−1, 1] are param-

x−µ Remark 1.302. Taking G (x) = e{−(1+γ( σ ))} , the cd f (1.1.849) can be expressed as

F (x; µ, σ, γ, λ) = (1 + λ) G (x) − λG (x)2 , which was discussed in Remark 1.28.

1.1.424

Transmuted Burr Type X (TBX)

The cd f and pd f of TBX are given, respectively, by

x ≥ 0, and

    2 φ 2 2φ F (x; θ, φ, λ) = (1 + λ) 1 − e−(θx) − λ 1 − e−(θx) ,

(1.1.851)

d F (x; θ, φ, λ), x > 0, (1.1.852) dx where θ > 0, φ > 0 and λ ∈ [−1, 1] are parameters.   2 φ Remark 1.303. Taking G (x) = 1 − e−(θx) , x ≥ 0 , the cd f (1.1.851) can be written as f (x; θ, φ, λ) =

F (x; θ, φ, λ) = (1 + λ) G (x) − λG (x)2 ,

which was discussed in Remark 1.28.

G. G. Hamedani

194

1.1.425

Odd Lindley Exponentiated Weibull (OLi-EW)

The cd f and pd f of OLi-EW are given, respectively, by

x ≥ 0, and

 n h i o β α 1 + 1 − 1 − e−x n F (x; α, β, 1) = 1 −   o × β α −x 2 1− 1−e  n h i o β α exp − 1 − 1 − e−x n ,  α o 1 − 1 − e−xβ

d F (x; α, β, 1), dx where α, β, a are all positive parameters. f (x; α, β, 1) =

x > 0,

(1.1.853)

(1.1.854)

Remarks 1.304. (a) The parameter a given by the author must h i be equal to 1, which is corβ

α

rected in the formula (1.1.853). (b) Taking G (x) = 1 − e−x , x ≥ 0 , the cd f (1.1.853) can be expressed as     1 1 G (x) F (x; α, β, 1) = 1 − 1+ exp − , x ≥ 0, 2 G (x) G (x) which is a special case of the cd f (1.1.87).

1.1.426

Transmuted Lomax-G (TL-G)

The cd f and pd f of TL-G are given, respectively, by F (x; α, η) = and


1 1 − (1 + G (x; η))−α , x ∈ R, −α 1−2

f (x; α, η) =

αg (x; η) (1 + G (x; η))−α−1 , 1 − 2−α

x ∈ R,

(1.1.855)

(1.1.856)

where α > 0, is a parameter and G (x; η) is a baseline cd f with the corresponding pd f g (x; η) which may depend on the parameter vector η.

1.1.427

Power Lindley Geometric (PLG)

The cd f and pd f of PLG are given, respectively, by   α α 1 − β+1+βx e−βx β+1 h  i , x ≥ 0, F (x; α, β, θ) = α −βxα 1 − θ β+1+βx e β+1

(1.1.857)

Introduction

195

and α

(1 − θ) αβ2 xα−1 (1 + xα )e−βx f (x; α, β, θ) =  io2 , n h α −βxα e (β + 1) 1 − θ β+1+βx β+1

x > 0,

(1.1.858)

where α, β, θ, are all positive parameters.

1.1.428

Odd Lomax Exponential (OLxEx)

The cd f and pd f of OLxEx are given, respectively, by h i−β F (x; α, β, λ) = 1 − αβ α − 1 + eλx , x ≥ 0,

and

d F (x; α, β, λ), dx where α, β, λ, are all positive parameters. f (x; α, β, λ) =

x > 0,

(1.1.859)

(1.1.860)

Remark 1.305. For a = 1 and p = 1 − α , the cd f (1.1.859) is a special case of the cd f (1.1.23).

1.1.429

Jamal Weibull-X (JW-X)

The cd f and pd f of JW-X are given, respectively, by − log (1 − F (x; η)a ) F (x; α, β, a, η) = 1 − exp −α 1 − F (x; η)a "



β #

, x ∈ R,

(1.1.861)

and d F (x; α, β, a, η), dx where α, β, a and η are all positive parameters. f (x; α, β, a, η) =

x ∈ R,

(1.1.862)

Remark 1.306. The cd f (1.1.861) is the same as the cd f (1.1.131).

1.1.430

Topp-Leone Exponentiated-G (TLEx-G)

The cd f and pd f of TLEx-G are given, respectively, by

and

n  2 oθ F (x; α, θ, ϕ) = 1 − 1 − H (x; ϕ)α , x ≥ 0,

d F (x; α, θ, ϕ), x > 0, dx where α, θ, ϕ are all positive parameters and H (x; ϕ) is a baseline cd f . f (x; α, θ, ϕ) =

(1.1.863)

(1.1.864)

G. G. Hamedani

196

Remark 1.307. Taking K (x; ϕ) = H (x; ϕ)α , x ≥ 0, the cd f (1.1.863) is a special case of the cd f of Rezaei et al. (2016) given in Remark 1.12.

1.1.431

New Extended Burr III (NEBIII)

The cd f and pd f of NEBIII are given, respectively, by
−αβ 1 + x−δ F (x; α, β, δ) =
−αβ n
−β oα , x ≥ 0, 1 + x−δ + 1 − 1 + x−δ

and

(1.1.865)

d F (x; α, β, δ), x > 0, (1.1.866) dx where α, β, δ are all positive parameters.
−β Remark 1.308. Taking G (x) = 1 + x−δ , x ≥ 0, the cd f (1.1.865) can be written as f (x; α, β, δ) =

G (x)α F (x; α, β, δ) = , x ≥ 0, G (x)α + G (x)α

which is a special case of the cd f discussed in Remark 1.189.

1.1.432

Zero Truncated Poisson Topp Leone Fréchet (ZTPTL-Fr)

The cd f and pd f of ZTPTL-Fr are given, respectively, by

F (x; α, β, δ, θ) =

    β x−β β x−β θ −θδ −δ 1 − exp −αe 2−e 1 − e−a

, x ≥ 0,

(1.1.867)

and d F (x; α, β, δ, θ), dx where α ∈ R, β > 0, δ > 0 and θ > 0 are parameters. f (x; α, β, δ, θ) =

x > 0,

(1.1.868)

Remark 1.309. The cd f (1.1.867) is a special case of the cd f (1.1.181).

1.1.433

Alpha Power Transformed Inverse Lindley (APTIL)

The cd f and pd f of APTIL are given, respectively, by F (x; α, λ) = and

α

h i λ 1+ (λ+1)x e−λ/x −1

α−1

, x ≥ 0,

(1.1.869)

Introduction

197

d F (x; α, λ), x > 0, (1.1.870) dx where α > 0, α 6= 1 and λ > 0 are parameters. i h λ Remark 1.310. Taking G (x) = 1 + (λ+1)x e−λ/x, x ≥ 0, the cd f (1.1.869) can be expressed as f (x; α, λ) =

F (x; α, λ) = which was discussed in Remark 1.241.

1.1.434

αG(x) − 1 , x ≥ 0, α−1

Beta Type I Generalized Half Logistic (BTIGHL)

The cd f and pd f of BTIGHL are given, respectively, by

F (x; δ, ζ, a, b, q) =

1 B (a, b)

Z 1−

2 x−ζ 1+e δ

!q

0

ua−1 (1 − u)b−1 , x ≥ ζ,

(1.1.871)

and d F (x; δ, ζ, a, b, q), x > ζ, (1.1.872) dx where δ, ζ, a, b, q are all positive parameters.  q 2 Remark 1.311. Taking G (x) = 1 − , x ≥ ζ, the cd f (1.1.871) can be written as x−ζ f (x; δ, ζ, a, b, q) =

1+e

δ

1 F (x; δ, ζ, a, b, q) = B (a, b)

Z G(x) 0

ua−1 (1 − u)b−1 , x ≥ ζ,

which has appeared in Remark 1.15.

1.1.435

Burr-Hatke Extended Burr XII (BHEBXII)

The cd f and pd f of BHEBXII are given, respectively, by

x ≥ 0, and

 h ib θ −β α 1 − 1 − (x + 1)   F (x; α, β, θ, b) = 1 − h ib  , −β α 1 − log 1 − 1 − (x + 1)

d F (x; α, β, θ, b), dx where α, β, θ, b are all positive parameters. f (x; α, β, θ, b) =

x > 0,

(1.1.873)

(1.1.874)

G. G. Hamedani

198

h ib Remark 1.312. Taking G (x) = 1 − (xα + 1)−β , x ≥ 0, the cd f (1.1.873) can be expressed as θ G (x)
, x ≥ 0, F (x; α, β, θ, b) = 1 − 1 − log G (x) 

which is the cd f (1.1.217).

1.1.436

Wrapped Lindley (WL)

The cd f and pd f of WL are given, respectively, by (1 + λ (1 + x)) e−λx 1− λ+1
2πλe−2πλ 1 − e−λx
2 , (λ + 1) 1 − e−2πλ

1 F (x; λ) = 1 − e−2πλ

0 ≤ x ≤ 2π, and

!

−

" # 1+x 2πe−2πλ λ2 e−λx + f (x; λ) =
2 , 0 < x < 2π, λ + 1 1 − e−2πλ 1 − e−2πλ

(1.1.875)

(1.1.876)

where λ > 0 is a parameter.

1.1.437

Truncated Cauchy Power-G (TCP-G)

The cd f and pd f of TCP-G are given, respectively, by F (x; α, ζ) = and f (x; α, ζ) =

  4 arctan G (x; ζ)α , π

4αg (x; ζ) G (x; ζ)α−1 1 + G (x; ζ)2α

,

x ∈ R,

(1.1.877)

x∈R,

(1.1.878)

where α > 0 is a parameter and G (x; ζ) is a baseline cd f with the corresponding pd f g (x; ζ) which may depend on the parameter vector ζ.

1.1.438

Slash Lindley-Weibull (SLW)

The cd f and pd f of SLW are given, respectively, by F (x; α, β, θ, q) = and

Z x 0

f (u; α, β, θ, q),

x ≥ 0,

(1.1.879)

Introduction

199

αθ2 xα−1 × βα (θ + 1) " !α # ( Z 1 1/q xt t α/q 1 + exp − β 0

f (x; α, β, θ, q) =

xt 1/q β

!α )

dt,

x > 0,

(1.1.880)

where α, β, θ, q are all positive parameters.

1.1.439

Risti´c-Balakrishnan Extended Exponential (RBEE)

The cd f and pd f of RBEE are given, respectively, by F (x; α, β, a) = 1 −

γ [a, ρ (x)] , Γ (a)

x ≥ 0,

(1.1.881)

and f (x; α, β, a) =

α2 (1 + βx) e−αxρ (x)α−1 , (α + β) Γ (a)

where α, β, a are all positive parameters, γ (a, z) = log[α + β − (α + β + αβx) e−αx ].

1.1.440

x > 0,

(1.1.882)

R z α−1 −t e dt and ρ (x) = log(α + β) − 0t

Exponentiated Generalized Standardized Gumbel (EGSGu)

The cd f and pd f of EGSGu are given, respectively, by

and

   a
b F (x; a, b) = 1 − 1 − exp −e−x ,

x ∈ R,

    a−1 f (x; a, b) = ab exp −x − e−x 1 − exp −e−x ×   −x  a
b−1 1 − 1 − exp −e ,

(1.1.883)

(1.1.884)

x ∈ R , where a > 0, b > 0 are parameters.

1.1.441

Unit-Lindley (unit-Lindley)

The cd f and pd f of unit-Lindley are given, respectively, by   θx θx e− 1−x , 0 ≤ x ≤ 1, F (x; θ) = 1 − 1 − (1 + θ) (x − 1) and θ2 (1 − x)−3 − θx f (x; θ) = e 1−x , 1+θ where θ > 0 is a parameter.

0 < x < 1,

(1.1.885)

(1.1.886)

G. G. Hamedani

200

1.1.442

Odd Log-Logistic Dagum (OLLDa)

The cd f and pd f of OLLDa are given, respectively, by

x ≥ 0, and

F (x; α, a, b, c) = h

1+

h 1+


i−αc x −b + a


i−αc x −b a

 h 1− 1+


i−c x −b a

α ,

(1.1.887)

d F (x; α, a, b, c), x > 0, (1.1.888) dx where α, a, b, c are all positive parameters. h
−b i−c Remark 1.313. Taking G (x) = 1 + ax , x ≥ 0, the cd f (1.1.887) can be expressed as f (x; α, a, b, c) =

F (x; α, a, b, c) =

G (x)α , x ≥ 0, G (x)α + G (x)α

which is a special case of the cd f discussed in Remark 1.39.

1.1.443

Type I Half-Logistic Modified Weibull (TIHLMW)

The cd f and pd f of TIHLMW are given, respectively, by 
 1 − exp −λ αx + θxβ 
 , x ≥ 0, F (x; α, β, λ, θ) = 1 + exp −λ αx + θxβ

(1.1.889)

and

d F (x; α, β, λ, θ), x > 0, (1.1.890) dx where α, β, λ, θ are all positive parameters. 
 Remark 1.314. Taking G (x) = 1 − exp −λ αx + θxβ , x ≥ 0, the cd f (1.1.889) can be written as f (x; α, β, λ, θ) =

F (x; α, β, λ, θ) = which is a special case of the cd f (1.1.439).

1 − G (x) , x ≥ 0, 1 + G (x)

Introduction

1.1.444

201

Odd Dagum (OddD-G)

The cd f and pd f of OddD-G are given, respectively, by F (x; a, b, θ, η) = 

and

G (x; η)ab G (x; η)a + θG (x; η)a

b , x ∈ R,

(1.1.891)

d F (x; a, b, θ, η), x ∈ R, (1.1.892) dx where a, b, θ are all positive parameters and G (x; η) is a baseline cd f which may depend on the parameter vector η. f (x; a, b, θ, η) =

Remark 1.315. The cd f (1.1.891) can be expressed as F (x; a, b, θ, η) = a

G (x; η)a   G (x; η)a + θG (x; η)a

!b

, x ∈ R.

The cd f K (x) = G(x;η)G(x;η) , x ∈ R, was discussed in Remark 1.120 and F (x; a, b, θ, η) a +θG(x;η)a is simply a power of K (x), so similar characterizations can be stated for the cd f (1.1.891). Please see cd f (1.1.317) as well.

1.1.445

Unit-Weibull (UW)

The cd f and pd f of UW are given, respectively, by h i F (x; α, β) = exp −α (− logx)β , 0 ≤ x ≤ 1,

(1.1.893)

and

h i f (x; α, β) = αβx−1 (− logx)β−1 exp −α (− log x)β ,

0 < x < 1,

(1.1.894)

where α > 0, β > 0 are parameters.

1.1.446

Generalized DUS Lindley (GDUSL)

The cd f and pd f of GDUSL are given, respectively, by
α  θx exp 1 − e−θx 1 + 1+θ −1 F (x; α, θ) = , x ≥ 0, e−1 and   α−1 αθ2 (1 + x) e−θx θx −θx f (x; α, θ) = 1−e 1+ × (e − 1) (θ + 1) 1+θ   α θx −θx exp 1 − e 1+ , 1+θ

(1.1.895)

(1.1.896)

G. G. Hamedani

202

x > 0, where α > 0, θ > 0 are parameters.

1.1.447

Poisson Rayleigh Burr XII (PRBXII)

The cd f and pd f of PRBXII are given, respectively, by    2 −[ −1+(xα +1)b ] 1 − exp −λ 1 − e , F (x; a, b, λ) = 1 − e−λ x ≥ 0, and d F (x; a, b, λ), dx where a, b, λ are all positive parameters. f (x; a, b, λ) =

Remark 1.316. Taking G (x) = 1 − e−[−1+(x pressed as F (x; a, b, λ) = which has appeared in Remark 1.40.

1.1.448

x > 0,

(1.1.897)

(1.1.898)

α +1)b 2

] , x ≥ 0, the cd f (1.1.897) can be ex-

1 − e−λG(x) , x ≥ 0, 1 − e−λ

Exponential-Gamma (EG)

The cd f and pd f of EG are given, respectively, by F (x; α, θ) = and f (x; α, θ) =

h   i 1 θ 1 − e−θx + θα T , x ≥ 0, θ + Γ (α)
θ θ + θα−1 xα−1 e−θx , θ + Γ (α)

x > 0,

(1.1.899)

(1.1.900)

where α, θ are positive parameters.

1.1.449

Extended Beta Power Function (EBPF)

The cd f and pd f of EBPF are given, respectively, by 1 F (x; α, θ, a, b) = B (a, b)

Z ( x )α+1 θ 0

t a−1 (1 − t)b−1 dt, 0 ≤ x ≤ θ,

(1.1.901)

and d F (x; α, θ, a, b), dx where α, θ, a, b are all positive parameters. f (x; α, θ, a, b) =

0 < x < θ,

(1.1.902)

Introduction Remark 1.317. Taking G (x) =


x α+1 , θ

1 F (x; α, θ, a, b) = B (a, b)

203

0 ≤ x ≤ θ, the cd f (1.1.901) can be written as

Z G(x) 0

t a−1 (1 − t)b−1 dt, 0 ≤ x ≤ θ,

which was discussed in Remark 1.96.

1.1.450

Arcsine Exponentiated-X (ASE-X)

The cd f and pd f of ASE-X are given, respectively, by F (x; α, η) = and


2 arcsin G (x; η)α , x ∈ R, π

(1.1.903)

d F (x; α, η) , x ∈ R, dx where α > 0, η ∈ R are parameters and G (x; η) is a baseline cd f . f (x; α, η) =

(1.1.904)

Remark 1.318. Taking K (x) = G (x; η)α , x ∈ R, the cd f (1.1.903) can be written as F (x; α, η) =

2 arcsin (K (x)), x ∈ R, π

which has appeared in Remark 1.229.

1.1.451

Lindley Quasi XGamma (LQXG)

The cd f and pd f of LQXG are given, respectively, by

F (x; α, θ) =

1

× 2 (θ + α)2 
  (θ + α) 2α + 2 − 2α + θ2 x2 + 2θx + 2 e−θx , +2 (θ − 1) {θ + α − (θ + αθx + α) e − θx}

x ≥ 0,

(1.1.905)

and f (x; α, θ) =

θe−θx (θ + α)2

x > 0, where α > 0, θ > 0 are parameters.

(

  ) 2 2 (θ + α) α + θ 2x

+θ (θ − 1) (1 + αx)

,

(1.1.906)

G. G. Hamedani

204

1.1.452

Type II Half Logistic Exponentiated Exponential (TIIHLEE)

The cd f and pd f of TIIHLEE are given, respectively, by

and


λ 2 1 − e−βx F (x; α, β, λ) = 
λ , 1 + 1 − e−βx

x ≥ 0,

(1.1.907)

d F (x; α, β, λ), x > 0, (1.1.908) dx where α, β, λ are all positive parameters. 
λ Remark 1.319. Taking G (x) = 1 − e−βx x ≥ 0, the cd f (1.1.907) can be expressed as f (x; α, β, λ) =

F (x; α, β, λ) =

2G (x) , x > 0, 1 + G (x)

which is a special case of the cd f (1.1.105).

1.1.453

Weighted Ishita (WI)

The cd f and pd f of WI are given, respectively, by F (x; θ, c) =

Z x 0

f (u; θ, c)du,

x ≥ 0,

(1.1.909)

and 2 −θx xc θ(c+3)(θ+x )e f (x; θ, c) = , c! (θ3 + (c + 1) (c + 2))

x > 0,

(1.1.910)

where θ > 0, c > 0 are parameters.

1.1.454

Quasi Sujatha (QS)

The cd f and pd f of QS are given, respectively, by   θx (θx + θ + 2) −θx F (x; θ, α) = 1 − 1 + e , x ≥ 0, αθ + θ + 2

(1.1.911)

and

d F (x; θ, α), dx where θ > 0, α > 0 are parameters. f (x; θ, α) =

x > 0,

Remark 1.320. The cd f (1.1.911) is a special case of the cd f (1.1.243).

(1.1.912)

Introduction

1.1.455

205

Weighted Nakagami (WN)

The cd f and pd f of WN are given, respectively, by   2 γ δ + θ/2, δxω F (x; δ, θ, ω) = , Γ (δ + θ/2)

x ≥ 0,

(1.1.913)

and f (x; δ, θ, ω) =

δx2 2δδ+θ/2 x2δ+β−1 e− ω , x > 0, δ+θ/2 Γ (δ + θ/2) ω

(1.1.914)

where δ, θ, ω are all positive parameters.

1.1.456

Weighted Version of Generalized Inverse Weibull (WVGIW)

The cd f and pd f of WVGIW are given, respectively, by   β Γ 1 − θ/β, λα xβ , F (x; α, β, λ, θ) = Γ (1 − θ/β)

x ≥ 0,

(1.1.915)

βαβ−θ λ1−θ/β θ−β−1 −λ(α/x)β x e , x > 0, Γ (1 − θ/β)

(1.1.916)

and

f (x; α, β, λ, θ) =

where α, β, λ, θ are all positive parameters.

1.1.457

Weighted Inverse Lévy (WIL)

The cd f and pd f of WIL are given, respectively, by
γ c + 12 , θx 2
F (x; c, θ) = , Γ c + 21

x ≥ 0,

(1.1.917)

and

 c+ 1 2 1 θx θ 1
xc− 2 e− 2 , x > 0, f (x; c, θ) = 1 2 Γ c+ 2

(1.1.918)

where c > 0 and θ > 0 are parameters.

1.1.458

Exponentiated Length Biased Exponential (ELBE)

The cd f and pd f of ELBE are given, respectively, by

and

n h x  −x/θ iα oβ F (x; α, β, θ) = 1 − 1 + e , x ≥ 0, θ

(1.1.919)

G. G. Hamedani

206

h x  −x/θ iα−1 e × f (x; α, β, θ) = αβθ−2xe−x/θ 1 + θ n h x  −x/θ iα oβ−1 1− 1+ e , θ

(1.1.920)

x > 0, where α, β and θ are all positive parameters.

1.1.459

Length-Biased Suja (LBS)

The cd f and pd f of LBS are given, respectively, by   2 α + 20x3 α2 120x + 60x

α e−αx + 1 + 5x4 α3 + xα4 1 + x4 F (x; α) = 1 − , 120 + α4 and f (x; α) = where α > 0 is a parameter.

1.1.460


α6 x + x5 e−αx , 4 120 + α

x > 0,

(1.1.921)

(1.1.922)

Alpha-Power Generalized Inverse Lindley (APGIL)

The cd f and pd f of APGIL are given, respectively, by F (x; α, θ, λ) =

α



−λ

x 1+ θ1+θ

and



−λ

e−θ x

α −1

−1

,

x ≥ 0,

(1.1.923)

d F (x; α, θ, λ), x > 0, (1.1.924) dx where α > 0, α 6= 1, θ > 0 and λ > 0 are parameters.   −λ x−λ Remark 1.321. Taking G (x) = 1 + θ1+θ e−θ x , x ≥ 0, the cd f (1.1.923) can be written as f (x; α, θ, λ) =

F (x; α, θ, λ) = which has appeared in Remark 1.34.

α G(x) − 1 , α −1

x ≥ 0,

Introduction

1.1.461

207

Exponentiated Cubic Transmuted Exponential (ECTE)

The cd f and pd f of ECTE are given, respectively, by  a F (x; a, m, β, λ) = 1 + (λ + β − 1) e−mx − (λ + 2β) e−2mx + βe−3mx , x ≥ 0, (1.1.925)

and

  f (x; a, m, β, λ) = ame−mx (1 − λ − β) + 2 (λ + 2β) e−mx − 3βe−2mx ×  a−1 1 + (λ + β − 1) e−mx − (λ + 2β) e−2mx + βe−3mx , x > 0,

(1.1.926)

where a > 0, m > 0, |β| ≤ 1, |λ| ≤ 1 and −2 ≤ λ + β ≤ 1 are parameters.

1.1.462

Transmuted Alpha Power Inverse Rayleigh (TAPIR)

The cd f and pd f of the corrected version TAPIR are given, respectively, by h  i  −λ/x2 −λ/x2 (α + β) − 1 + βαe α − αe , F (x; α, β, λ) = 1 − (α − 1)2

(1.1.927)

x ≥ 0, and

f (x; α, β, λ) =

2λ ln (α)e

−λ/x2

(α − 1)2 x3

αe

−λ/x2

×

  −λ/x2 β (α + 1) + (α − 1) − 2βαe ,

(1.1.928)

x > 0, where α > 0, α 6= 1, λ > 0 and |β| ≤ 1 are parameters.

Remark 1.322. The cd f s given by Malik and Ahmad (2019) for both cases α = 1 and α 6= 1 are incorrect. The cd f (1.1.927) is the corrected version for α 6= 1.

1.1.463

New Beta Power Transformed (NBPT)

The cd f and pd f of NBPT are given, respectively, by F (x; β, η) =

βG(x;η) − G (x; η) , β

x ∈ R,

(1.1.929)

and f (x; β, η) =

i g (x; η) h 1 + (logβ)βG(x;η) , x ∈ R, β

where β > 0, β 6= 1 and η > 0 are parameters.

(1.1.930)

G. G. Hamedani

208

1.1.464

Generalized Gamma Exponentiated Weibull (GGEW)

The cd f and pd f of GGEW are given, respectively, by h i p    α γ dp , − α1 ln 1 − e−(x/β)   , F (x; α, β, d, p) = 1 − Γ dp

x ≥ 0,

(1.1.931)

and

d F (x; α, β, d, p), x > 0, (1.1.932) dx where α, β, d, p are all positive parameters and γ (·, ·) is the incomplete gamma function. f (x; α, β, d, p) =

Remark 1.323. The cd f (1.1.499) is given as    β γ a, − ln 1 − e−αx , x ≥ 0. F (x; α, β, a) = 1 − Γ (a) The characterizations of the cd f (1.1.931) are similar to those of the cd f (1.1.499).

1.1.465

Marshall-Olkin Exponential Gompertz (MOEGo)

The cd f and pd f of MOEGo are given, respectively, by   − θ (eβx −1) −λ 1−e β

1−e

F (x; α, β, θ, λ) =

x ≥ 0, and

  , − θ (eβx −1) −λ 1−e β



α 1 − e−λ + α 1 − e




d F (x; α, β, θ, λ), x > 0, dx where α, β, θ, λ are all positive parameters and α = 1 − α. f (x; α, β, θ, λ) =

  − θ eβx −1  −λ1−e β

(

Remark 1.324. Taking G (x) = pressed as

1−e

1−e−λ

F (x; α, β, θ, λ) =

(1.1.933)

(1.1.934)

)

, x ≥ 0, the cd f (1.1.933) can be ex-

G (x) , x ≥ 0, G (x) + αG (x)

which is a special case of the cd f mentioned in Remark 1.39.

Introduction

1.1.466

209

Exponentiated Truncated Inverse Weibull-Generated (ETIW-G)

The cd f and pd f of ETIW-G are given, respectively, by   −b a F (x; a, b, ζ) = 1 − e1−(1−G(x;ζ)) ,

x ∈ R,

(1.1.935)

and

d F (x; a, b, ζ), x ∈ R, (1.1.936) dx where a > 0, b > 0 are parameters and and G (x; ζ) , x ∈ R, is a baseline cd f which may depend on the parameter vector ζ.
b Remark 1.325. Taking K (x) = 1 − G (x; ζ) , x ∈ R, the cd f (1.1.935) can be written as  a − K(x) F (x; a, b, ζ) = 1 − e K(x) , x ∈ R, f (x; a, b, ζ) =

which has been characterized in Hamedani (2019).

1.1.467

Lindley Rayleigh (LR)

The cd f and pd f of LR are given, respectively, by   −θx2 θx2 2σ2 , F (x; θ, σ) = 1 − 1 + e (θ + 1) 2σ2

x ≥ 0,

(1.1.937)

and

f (x; θ, σ) =

d F (x; θ, σ), dx

x > 0,

(1.1.938)

where θ > 0, σ > 0 are parameters. Remark 1.326. Ghitany et al. (2013) introduced the following distribution   β α F (x; α, β) = 1 − 1 + x exp [−βxα ] , x ≥ 0, β+1

which is more general than cd f (1.1.937). Please see Remark 1.22 as well.

1.1.468

Generalized Weibull Uniform (GWU)

The cd f and pd f of GWU are given, respectively, by n  oω −υ − log 1− ϕx

and

F (x; υ, ϕ, ω) = 1 − e

,

0 ≤ x ≤ ϕ,

d F (x; υ, ϕ, ω) , 0 < x < ϕ, dx where υ, ϕ, ω are all positive parameters. f (x; υ, ϕ, ω) =

(1.1.939)

(1.1.940)

G. G. Hamedani

210

Remark 1.327. Taking G (x) = ϕx , 0 ≤ x ≤ ϕ, the cd f (1.1.939) can be written as ω

F (x; υ, ϕ, ω) = 1 − e−υ{− log(1−G(x))} ,

0 ≤ x ≤ ϕ,

which has been discussed in Remark 1.50.

1.1.469

Burr XII Uniform (BXIIU)

The cd f and pd f of BXIIU are given, respectively, by

0 ≤ x ≤ θ, and

h n  x oc i−k , F (x; θ, c, k) = 1 − 1 + − log 1 − θ

d F (x; θ, c, k), 0 < x < θ, dx where θ, c, k are all positive parameters. f (x; θ, c, k) =

(1.1.941)

(1.1.942)

Remark 1.328. Taking G (x) = θx , 0 ≤ x ≤ θ, the cd f (1.1.941) can be written as  
c −k F (x; θ, c, k) = 1 − 1 + − log G (x) ,

which has been characterized before.

1.1.470

0 ≤ x ≤ θ,

G-Fixed-Topp-Leone (G-FTL)

The cd f and pd f of G-FTL are given, respectively, by F (x; λ) =

Z

λ G(x) 1 − λG(x)

−∞

v (t)dt = V

and f (x; λ) = 

λ G (x) 1 − λG (x)

λ g (x) 1 − λG (x)

2 v

!

, x ∈ R,

λ G (x) 1 − λG (x)

!

,

(1.1.943)

(1.1.944)

x ∈ R, where λ > 0 is a parameter, λ = 1 − λ and G (x) is a baseline cd f with the corresponding pd f g (x).

1.1.471

Generalized New Extended Weibull (GNEW)

The cd f and pd f of GNEW are given, respectively, by    βxα − σ2 2 x F (x; α, β, σ, θ) = 1 − exp −θx − e , x ≥ 0, and

(1.1.945)

Introduction

211

     2σ βxα − σ2 α−1 x × f (x; α, β, σ, θ) = 2θx + αβx + 3 e x    βxα − σ2 x , exp −θx2 − e

(1.1.946)

x > 0, where α, β, σ, θ are all positive parameters.

1.1.472

New Exponential Trigonometric (NET)

The cd f and pd f of NET are given, respectively, by
β2 + λ2 − αλ2 cos (βx) + αβλ sin(βx) e−λx F (x; α, β, λ) = 1 − , x ≥ 0, β2 + (1 − α) λ2

(1.1.947)

and
λ β2 + λ2 )(1 − α cos(βx) e−λx f (x; α, β, λ) = , β2 + (1 − α) λ2

x > 0,

(1.1.948)

where α ∈ (−1, 1), β ≥ 0 and λ > 0 are parameters.

1.1.473

Power Log-Dagum (PLD)

The cd f and pd f of PLD are given, respectively, by

and

n o−ζ υ F (x; υ, ζ, η) = 1 + e−(υx+sign(x)(η/υ)|x| ) , x ∈ R,   υ ζ υ + η |x|υ−1 e−(υx+sign(x)(η/υ)|x| ) f (x; υ, ζ, η) = , x ∈ R,  ζ+1 1 + e−(υx+sign(x)(η/υ)|x|)

(1.1.949)

(1.1.950)

where υ ∈ R, ζ > 0 and η ≥ 0 are parameters.

1.1.474

Poly-Exponential Transformation (PET)

The cd f and pd f of PET are given, respectively, by F (x; λ) =

h i 1 λG(x) 1 − (1 − λG (x)) e , 1 − (1 − λ) eλ

(1.1.951)

f (x; λ) =

λ2 g (x) G (x) eλG(x), 1 − (1 − λ) eλ

(1.1.952)

x ∈ R, and

where λ > 0 is a parameter.

x ∈ R,

G. G. Hamedani

212

1.1.475

New Weighted Transmuted Exponential (NWTE)

The cd f and pd f of NWTE are given, respectively, by

F (x; γ, λ, θ) =

h i
1 γ 1 − θ + θe−λx e−λx − 1 (1 + γ) γ +1 1 − 1+γ 1

(1 + γ) γ +1 − 1

, x ≥ 0,

(1.1.953)

x ≥ 0, and 1

f (x; γ, λ, θ) =

(1 + γ) γ +1 1

(1 + γ) γ +1 − 1

!

×


1 − θ + 2θe−λx e−λx h i− 1γ ,
γ −λx −λx 1 − 1+γ 1 − θ + θe e

(1.1.954)

x > 0, where γ > 0, λ > 0 and θ ∈ (−1, 1) are parameters.

1.1.476

Kumarswamy Poisson-G (KwP-G)

The cd f and pd f of KwP-G are given, respectively, by (

"

1 − e−λG(x) F (x; a, b, λ) = 1 − 1 − 1 − e−λ

#a )b

, x ∈ R,

(1.1.955)

and f (x; a, b, λ) =

d F (x; a, b, λ), dx

x ∈ R,

(1.1.956)

where a, b, λ are all positive parameters. Remark 1.329. Rezaei et al. (2016) proposed the following distribution n o b θ F (x; a, b, θ, ξ) = 1 − 1 − {1 − (1 − G (x; ξ))a } , x ∈ R.

Taking K (x) =

1−e−λG(x) , 1−e−λ

x ∈ R, the cd f (1.1.955) can be expressed as b

F (x; a, b, λ) = 1 − {1 − K (x)a } , x ∈ R, which is a special case of the cd f F (x; a, b, θ, ξ). Please see Remark 1.21 as well.

Introduction

1.1.477

213

Truncated Burr-G (TB-G)

The cd f and pd f of TB-G are given, respectively, by F (x; c, k) = and f (x; c, k) =

1 − [1 + G (x; η)c ] 1 − 2−k

−k

, x ∈ R,

ckg (x; η) G (x; η)c−1 [1 + G (x; η)c ] 1 − 2−k

−k−1

(1.1.957)

,

(1.1.958)

x ∈ R, where c > 0, k > 0 are parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η) which may depend on a parameter vector η.

1.1.478

Alpha Power Transformation Lomax (APTL)

The cd f and pd f of APTL are given, respectively, by λ

and

α1−( λ+x ) − 1 , F (x; α, β, λ) = α−1

x ≥ 0,

(1.1.959)

d F (x; α, β, λ), x > 0, (1.1.960) dx where α > 0, α 6= 1, β > 0 and λ > 0 are parameters.   λ Remark 1.330. Taking G (x) = 1 − λ+x , x ≥ 0, the cd f (1.1.959) can be written as f (x; α, β, λ) =

F (x; α, β, λ) = which has been discussed in Remark 1.31.

1.1.479

αG(x) − 1 , α−1

x ≥ 0,

Weighted Inverted Weibull (WIW)

The cd f and pd f of WIW are given, respectively, by   γ 1 − βc , xθβ
, F (x; β, θ, c) = Γ 1 − θc

x ≥ 0,

(1.1.961)

x > 0,

(1.1.962)

and

d F (x; β, θ, c), dx where β > 0, θ > 0 and c < β are parameters. f (x; β, θ, c) =

Remark 1.331. The cd f (1.1.961) is the same as the cd f (1.1.915).

G. G. Hamedani

214

1.1.480

Odd Exponential-Pareto IV (OEPIV)

The cd f and pd f of OEPIV are given, respectively, by     x 1/a α F (x; λ, a, θ, α) = 1 − exp −λ 1 + − 1 , x ≥ 0, θ and d F (x; λ, a, θ, α), dx where λ, a, θ, α are all positive parameters. f (x; λ, a, θ, α) =

x > 0,

(1.1.963)

(1.1.964)

Remark 1.332. The cd f (1.1.963) is similar to the cd f s (1.1.17), (1.1.59) and (1.1.649).

T-Dagum{Y } (TD{Y })

1.1.481

The cd f and pd f of TD{Y } are given, respectively, by   "  −α #−β   x , x ≥ 0, F (x; α, β, γ) = FT QY 1 +   γ

(1.1.965)

and

  

  

(x/γ)αβ−1 h   iβ+1  ×   1+ x α  γ (  )  −α −β fT QY 1 + xγ

f (x; α, β, γ) = αβγ

fY

),   −α −β QY 1 + xγ

(

x > 0,

(1.1.966)

where α, β, γ are all positive parameters, FT is the cd f of a random variable T , QY (·) is quantile function of a random variable Y and FR is a cd f of a random variable R.

1.1.482

Generalized Odd Inverted Exponential-G (GOIE-G)

The cd f and pd f of GOIE-G are given, respectively, by F (x; λ, γ, η) = exp {−λ [1 − G (x; η)] [1 + γG (x; η)] / G (x; η)} ,

(1.1.967)

x ∈ R, and

f (x; λ, γ, η) =

h i λg (x; η) 1 + γG (x; η)2

× G (x; η)2 exp {−λ [1 − G (x; η)] [1 + γG (x; η)] / G (x; η)} ,

(1.1.968)

Introduction

215

where λ > 0, γ ≥ −1 are parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η) which may depend on the parameter vector η.

1.1.483

Marshall-Olkin Burr Exponential-2 (MOBE-2)

The cd f and pd f of MOBE-2 are given, respectively, by   βαx e−αx 1 − 1 + 2−β   , F (x; α, β, γ) = βαx 1 − γ 1 + 2−β e−αx

x ≥ 0,

(1.1.969)

x ∈ R, and

d F (x; α, β, γ), x > 0, (1.1.970) dx where α > 0, 0 ≤ β ≤ 1 and γ > 0 are parameters and γ = 1 − γ.   βαx Remark 1.333. Taking G (x) = 1 − 1 + 2−β e−αx, x ≥ 0 , the cd f (1.1.969) can be expressed as f (x; α, β, γ) =

F (x; α, β, γ) =

G (x) , x ≥ 0, γ + (1 − γ) G (x)

which is a special case of the cd f discussed in Remark 1.315.

1.1.484

Exponentiated Power Generalized Weibull Power Series (EPGWPS)

The cd f and pd f of EPGWPS are given, respectively, by  h i  µ )α β 1−(1+λx C θ 1−e F (x; α, β, θ, λ, µ) = , C (θ)

x ≥ 0,

(1.1.971)

and d F (x; α, β, θ, λ, µ), x > 0, (1.1.972) dx n 0 where α, β, θ, λ, µ are all positive parameters and C (θ) = ∑∞ n=1 an θ , ai s are nonnegative real numbers. f (x; α, β, θ, λ, µ) =

Remark 1.334. The cd f given in (1.1.971) is a special case of the general form of
C θt (x)α F (x) = , x ≥ 0, C (θ) µ α

mentioned in Tahmasebi and Jafari (2016). Taking t (x) = 1−e1−(1+λx ) in the above F (x), yields (1.1.971). The distribution of Tahmasebi and Jafari (2016) has been characterized in Hamedani and Safavimanesh (2017b).

G. G. Hamedani

216

1.1.485

Exponentiated Transmuted Weibull Geometric (ETWG)

The cd f and pd f of ETWG are given, respectively, by     β α β α −( σx ) −( σx ) 1 + λe 1−e     , x ≥ 0, F (x; α, β, σ, λ, p) = β α β α −( σx ) −( σx ) 1− p+ p 1−e 1 + λe

(1.1.973)

and d F (x; α, β, σ, λ, p), x > 0, (1.1.974) dx where α > 0, β > 0, σ > 0, λ ∈ [−1, 1] and p ∈ (0, 1) are parameters.     β α β α −( σx ) −( σx ) 1 + λe , x ≥ 0, the cd f (1.1.973) Remark 1.335. Taking G (x) = 1 − e f (x; α, β, σ, λ, p) =

can be expressed as

F (x; α, β, σ, λ, p) = which was discussed in Remark 1.53.

1.1.486

G (x) , x ≥ 0, 1 − pG (x)

Complementary Exponential Geometric (CEG)

The cd f and pd f of ETWG are given, respectively, by   λe−βx A 1−p 1−e−βx ( ) F (x; β, λ, p) = 1 − , A (λ)

x ≥ 0,

(1.1.975)

and d F (x; β, λ, p), x > 0, (1.1.976) dx n where β > 0, λ > 0 , p ∈ (0, 1) are parameters and A (λ) = ∑∞ n=1 an λ is finite and {an }n≥1 is a sequence of positive real numbers. f (x; β, λ, p) =

−βx

Remark 1.336. Taking G (x) = 1 − 1−p e1−e−βx , x ≥ 0, the cd f (1.1.975) can be written as ( ) F (x; β, λ, p) = 1 − which has appeared in Remark 1.36.

A (λ [1 − G (x)]) , A (λ)

x ≥ 0,

Introduction

1.1.487

217

Complementary Exponentiated Lomax-Poisson (CELP)

The cd f and pd f of CELP are given, respectively, by h
θ i exp λ 1 − (1 + βx)−α −1 , F (x; α, β, λ, θ) = eλ − 1 and

x ≥ 0,

(1.1.977)

d F (x; β, λ, p), x > 0, (1.1.978) dx n where β > 0, λ > 0 , p ∈ (0, 1) are parameters and A (λ) = ∑∞ n=1 an λ is finite and {an }n≥1 is a sequence of positive real numbers.
θ Remark 1.337. Taking G (x) = 1 − (1 + βx)−α , x ≥ 0, the cd f (1.1.977) can be expressed as f (x; β, λ, p) =

exp [λG (x)] − 1 , eλ − 1 which is a special case of the cd f given in Remark 1.73. F (x; α, β, λ, θ) =

1.1.488

x ≥ 0,

Alpha Power Transformed Log-Logistic (APTLL)

The cd f and pd f of APTLL are given, respectively, by b −1

and

α1−(1+(x/a) ) − 1 F (x; α, a, b) = , α−1

x ≥ 0,

(1.1.979)

d F (x; α, a, b), x > 0, (1.1.980) dx where α (6= 1), a, b are all positive parameters.  −1 Remark 1.338. Taking G (x) = 1− 1 + (x/a)b , x ≥ 0, the cd f (1.1.979) can be written as f (x; α, a, b) =

F (x; α, a, b) = which was discussed in Remark 1.31.

1.1.489

αG(x) − 1 , α−1

x ≥ 0,

T-Kumarswamy (T-K)

The cd f and pd f of T-K are given, respectively, by n h io F (x; α, β) = FT QY 1 − (1 − xα )β , 0 ≤ x ≤ 1,

and

(1.1.981)

G. G. Hamedani

218

β−1

f (x; α, β) = αβxα−1 (1 − xα )

n h io fT QY 1 − (1 − xα )β n h io , fY QY 1 − (1 − xα )β

(1.1.982)

0 < x < 1, where α, β are positive parameters, FT is the cd f of a random variable T , QY (·) is quantile function of a random variable Y .

1.1.490

Zografos Balakrishnan Power Lindley (ZB-PL)

The cd f and pd f of ZB-PL are given, respectively, by     βxα + βxα γ a, − log 1 + β+1 F (x; α, β, a) = , x ≥ 0, Γ (a) and

(1.1.983)

α

αβ2 xα−1 (1 + xα ) e−βx × f (x; α, β, a) = Γ (a) (β + 1)    a−1 βxα α − log 1 + + βx , β+1 x > 0, where α, β, a are all positive parameters, γ (a, w) = R ∞ a−1 −u e du. 0 u

1.1.491

(1.1.984)

R w a−1 −u e du and Γ (a) = 0 u

Poisson Exponential-G (PE-G)

The cd f and pd f of PE-G are given, respectively, by λ

and

e−θ(1−G(x;φ)) − e−θ F (x; λ, θ, φ) = , x ∈ R, 1 − e−θ

d F (x; λ, θ, φ), x ∈ R, dx where λ, θ, φ are all positive parameters and G (x; φ) is a baseline cd f . f (x; α, β, a) =

(1.1.985)

(1.1.986)

Remark 1.339. The cd f (1.1.985) can be written as F (x; λ, θ, φ) =

h i θ 1−(1−G(x;φ))λ −1

e

eθ − 1

, x ∈ R.

Taking K (x) = 1 − (1 − G (x; φ))λ , x ∈ R, the cd f (1.1.985) will be a special case of the following cd f (for α = 1) ( ! ) 1 β (G (x))α F (x; α, β) = β exp
α − 1 , x ∈ R, e −1 (G (x))α + G (x) which discussed in Remark 1.73.

Introduction

1.1.492

219

Perturbed Half-Normal (PHN)

The cd f and pd f of PHN are given, respectively, by Z x

F (x; ζ, η, λ) =

ζ

f (u; ζ, η, λ) du, x ≥ ζ,

(1.1.987)

and f (x; ζ, η, λ) =

pπ 2

η arctan (λ)

− (x−ζ) 2

e

2η

2

    λ (x − ζ) 2Φ −1 , η

(1.1.988)

x > ζ , where ζ ∈ R, η > 0, λ > 0 are parameters and Φ (x) is the cd f of the standard normal random variable. Remark 1.340. WLOG we assume ζ = 0 and η = 1.

1.1.493

Normal-Poisson (NP)

The cd f and pd f of NP are given, respectively, by

and

x−µ 1 − e−θΦ( σ ) , x ∈ R, F (x; σ, µ, θ) = 1 − e−θ

(1.1.989)

d F (x; σ, µ, θ), x ∈ R, (1.1.990) dx where σ > 0, µ ∈ R, θ > 0 are parameters and Φ (x) is the cd f of the standard normal random variable.
Remark 1.341. Taking G (x) = Φ x−µ , x ∈ R, the cd f (1.1.989) can be expressed as σ f (x; σ, µ, θ) =

F (x; σ, µ, θ) =

which was discussed in Remark 1.40.

1.1.494

1 − e−θG(x) , x ∈ R, 1 − e−θ

Extended Exponentiated Weibull (EEW)

The cd f and pd f of EEW, for β < 0, are given, respectively, by h iα F (x; α, β, λ) = 1 − (1 − βλx)1/β , x ≥ 0,

(1.1.991)

and

f (x; α, β, λ) = αλ (1 − βλx)

1/β−1

where α > 0, β < 0, λ > 0 are parameters.

h

1 − (1 − βλx)

1/β

iα−1

, x > 0,

(1.1.992)

G. G. Hamedani

220

1.1.495

New Family of Heavy Tailed (NFHT)

The cd f and pd f of NFHT are given, respectively, by F (x; σ, ζ) = and

σG (x; ζ) , x ∈ R, σ − 1 + G (x; ζ)

d F (x; σ, ζ) , x ∈ R, dx where σ > 1, ζ ∈ R are parameters and G (x; ζ) is a baseline cd f . f (x; σ, ζ) =

(1.1.993)

(1.1.994)

Remark 1.342. The cd f (1.1.993) can be written as F (x; σ, ζ) =

G (x; ζ) , x ∈ R, 1 − γG (x; ζ)

where γ =

1 σ

1.1.496

New Heavy Tailed Family of Claim (NHTFC)

, which was mentioned in Remark 1.195. Please see the cd f (1.1.63) as well.

The cd f and pd f of NHTFC are given, respectively, by F (x; σ, ζ) = 1 − and

σG (x; ζ) , x ∈ R, σ − G (x; ζ)

d F (x; σ, ζ) , x ∈ R, dx where σ > 1, ζ > 0 are parameters and G (x; ζ) is a baseline cd f . f (x; σ, ζ) =

(1.1.995)

(1.1.996)

Remark 1.343. The cd f (1.1.995) can be written as F (x; σ, ζ) =

G (x; ζ) , x ∈ R, 1 + γG (x; ζ)

where γ =

1 σ

1.1.497

New Beta Power Transformed (NBPT)

, which is similar to the cd f in Remark 1.342.

The cd f and pd f of NBPT are given, respectively, by F (x; β, ζ) =

βG(x;ζ) − [1 − G (x; ζ)] , x ∈ R, β

(1.1.997)

and i g (x; ζ) h 1 + (log β)βG(x;ζ) , x ∈ R, (1.1.998) β where β > 0, β 6= 1 and ζ > 0 are parameters and G (x; ζ) is a baseline cd f with the corresponding pd f g (x; ζ). f (x; β, ζ) =

Introduction

1.1.498

221

Transmuted Type II Generalized Logistic (TTIIGL)

The cd f and pd f of TTIIGL are given, respectively, by

F (x; λ, b) =

(1 + e−x )

b

n

o b (1 + e−x ) − (1 − λ) e−bx − λe−2bx (1 + e−x )2b

x ∈ R,

(1.1.999)

and f (x; λ, b) =

d F (x; λ, b), x ∈ R, dx

(1.1.1000)

where b > 0, |λ| ≤ 1 are parameters. Remark 1.344. Taking G (x) = 1 −

e−bx , (1+e−x)b

x ∈ R, the cd f (1.1.999) can be expressed as

F (x; λ, b) = (1 + λ) G (x) − λG (x)2 , x ∈ R, which is a special case of the cd f discussed in Remark 1.110.

1.1.499

Beta-Complementary Exponential Power Series (BCEPS)

The cd f and pd f of BCEPS are given, respectively, by

1 F (x; a, b, β, θ) = B (a, b)

−βx Z C(θ(1−e )) C(θ)

0

wa−1 (1 − w)b−1 dw, x ≥ 0,

(1.1.1001)

and

f (x; a, b, β, θ) =

θβe−βx × B (a, b)

C0 θ 1 − e−βx





a−1 

b−1 C θ 1 − e−βx C (θ) −C θ 1 − e−βx C (θ)a+b−1 F (x; a, b, β, θ)

x > 0,

,

(1.1.1002)

n where a, b, β, θ are all positive parameters and C (θ) ∑∞ n=1 an θ , an ≥ 0, and θ ∈ (0, s).

1.1.500

Zubair-G (Zubair-G)

The cd f and pd f of Zubair-G are given, respectively, by 2

and

eαG(x;ζ) − 1 F (x; α, ζ) = , eα − 1

x ∈ R,

(1.1.1003)

G. G. Hamedani

222

d F (x; α, ζ) , x ∈ R, (1.1.1004) dx where α > 0, ζ > 0 are parameters and G (x; ζ) is a baseline cd f which may depends on the parameter ζ. f (x; α, ζ) =

Remarks 1.345. a) Taking β = eα and K (x) = G (x; ζ)2 , x ∈ R, the cd f (1.1.1003) can be written as F (x; α, ζ) =

βK(x) − 1 , β−1

x ∈ R,

which was discussed in Remark 1.34. b) The power 2 in the cd f (1.1.1003) can be replaced with γ > 0.

1.1.501

Generalized Class (GC)

The cd f and pd f of GC are given, respectively, by F (x; α, λ, ζ) = (1 + λ) G (x; ζ) − λG (x; ζ)α , x ∈ R,

(1.1.1005)

and d F (x; α, λ, ζ) , x ∈ R, (1.1.1006) dx where α > 0, ζ > 0, |λ| ≤ 1 are parameters and G (x; ζ) is a baseline cd f which may depends on the parameter ζ. f (x; α, λ, ζ) =

Remarks 1.346. a) The cd f (1.1.1005) is a special case of the cd f (1.1.179). b) It is also a special case of the distribution mentioned in Remark 1.110.

1.1.502

Odd Lindley Half Logistics (OLiHL)

The cd f and pd f of OLiHL are given, respectively, by F (x; λ) = 1 −

λex + λ + 2 − λ (ex −1) e 2 , 2 (λ + 1)

x ≥ 0,

(1.1.1007)

and f (x; λ) =

d F (x; λ) , dx

x > 0,

(1.1.1008)

where λ > 0 is a parameter. Remark 1.347. Taking G (x) =

1−e−x 1+e−x ,

F (x; λ) = 1 −

x ≥ 0, the cd f (1.1.1007) can be expressed as

G(x) λ + G (x) −λ G(x) e , (1 + λ) G (x)

which is exactly the cd f mentioned in Remark 1.126.

x ≥ 0,

Introduction

1.1.503

223

Modified Beta Generalized Linear Failure Rate (MBGLFR)

The cd f and pd f of MBGLFR are given, respectively, by 1 F (x; α, λ, θ, a, b, c) = B (a, b)

Z Q(x) 0

t a−1 (1 − t)b−1 dt,

x ≥ 0,

(1.1.1009)

and f (x; α, λ, θ, a, b, c) =

d F (x; α, λ, θ, a, b, c), x > 0, dx

where α, λ, θ, a, b, c are all positive parameters and Q (x) = which is a cd f .

(1.1.1010)

 α − λx+ θ x2 c 1−e ( 2 )  α, − λx+ θ x2 1−(1−c) 1−e ( 2 )

x ≥ 0,

Remark 1.348. The cd f (1.1.1009) for a given cd f Q (x) has been discussed in Remark 1.15.

1.1.504

Truncated Weibull Fréchet (TWFr)

The cd f and pd f of TWFr are given, respectively, by h i δ 1 − exp −αe−β(µ/x) F (x; α, β, δ, µ) = , 1 − e−α and

x ≥ 0,

d F (x; α, β, δ, µ), dx where α, β, δ, µ are all positive parameters. f (x; α, β, δ, µ) =

x > 0,

(1.1.1011)

(1.1.1012)

δ

Remark 1.349. Taking G (x) = e−β(µ/x) , x ≥ 0, the cd f (1.1.1011) can be expressed as F (x; α, β, δ, µ) = which appeared in Remark 1.146.

1.1.505

1 − exp [−αG (x)] , 1 − e−α

x ≥ 0,

Beta Poisson-G (BP-G)

The cd f and pd f of BP-G are given, respectively, by 1 F (x; m, n, λ) = B (m, n)

Z

1−e−λG(x) 1−e−λ

0

t m−1 (1 − t)n−1 dt,

x ∈ R,

(1.1.1013)

and d F (x; m, n, λ), x ∈ R, dx where m, n, λ are all positive parameters and G (x) is a baseline cd f . f (x; m, n, λ) =

(1.1.1014)

G. G. Hamedani

224

1−e−λG(x) , 1−e−λ

Remark 1.350. Taking Q (x) =

1 F (x; m, n, λ) = B (m, n) which appeared in Remark 1.15.

1.1.506

x ∈ R, the cd f (1.1.1013) can be expressed as

Z Q(x) 0

t m−1 (1 − t)n−1 dt, ,

x ∈ R,

Topp-Leone Marshall-Olkin-G (TLMO-G)

The cd f and pd f of TLMO-G are given, respectively, by 

b

2

δ2 G (x; η)   F (x; δ, b, η) = 1 − h i2  , 1 − δG (x; η)

and



2bδ2 g (x; η) G (x; η)

 f (x; δ, b, η) =  h

1 − δG (x; η)



2

x ∈ R,

(1.1.1015)

x ∈ R,

(1.1.1016)



 i3  × b−1

δ2 G (x; η)   1 − h i2  1 − δG(x; η)

,

where δ > 0, b > 0, δ = 1 − δ are parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η) , which may depend on a parameter vector η.

1.1.507

Beta Modified Weibull Power Series (BMWPS)

The cd f and pd f of BMWPS are given, respectively, by

F (x; α, β, σ, θ, a, b, k) = 1 −

 
k  C θ 1 − I1−e−αx−βxσ (a, b) C (θ)

, x ≥ 0,

(1.1.1017)

and d F (x; α, β, σ, θ, a, b, k), (1.1.1018) dx x > 0, where σ > 0, θ > 0, a > 0, b > 0, k > 0, α ≥ 0, β ≥ 0, α + β > 0 are parameters and n C (θ) = ∑∞ n=1 an θ is finite and {an }n≥1 is a sequence of positive real numbers.
k Remark 1.351. Taking G (x) = I1−e−αx−βxσ (a, b) , x ≥ 0, the cd f (1.1.1017) can be written as f (x; α, β, σ, θ, a, b, k) =

F (x; α, β, σ, θ, a, b, k) = 1 − which appeared in Remark 1.36.

C (θ (1 − G (x))) , x ≥ 0, C (θ)

Introduction

1.1.508

225

Fréchet Weibull (FW)

The cd f and pd f of FW are given, respectively, by "  αk # α λ , x ≥ 0, F (x; α, β, λ, k) = exp −β x

(1.1.1019)

and d F (x; α, β, λ, k), x > 0, dx where α, β, λ, k are all positive parameters. f (x; α, β, λ, k) =

(1.1.1020)

Remark 1.352. The cd f (1.1.1019) is a special case of the cd f s which appeared in Remarks 1.13 and 1.19.

1.1.509

Odd Lomax Fréchet (OLxF)

The cd f and pd f of OLxF are given, respectively, by F (x; α, β, a, b) = 1 − β

α

(

and

a

e−( x )b β+ a 1 − e−( x )b

)−α

,

x ≥ 0,

d F (x; α, β, a, b), x > 0, dx where α, β, a, b are all positive parameters. f (x; α, β, a, b) =

(1.1.1021)

(1.1.1022)

a

Remark 1.353. Taking G (x) = e−( x )b , x ≥ 0, the cd f (1.1.1021) can be written as  F (x; α, β, a, b) = 1 − β β + α

which is exactly the cd f (1.1.57).

1.1.510

G (x) 1 − G (x)

−α

, x ≥ 0,

Burr X Weibull (BXW)

The cd f and pd f of BXW are given, respectively, by

and

  h i2 θ xβ F (x; θ, β) = 1 − exp − e − 1 , x ≥ 0, f (x; θ, β) =

where θ, β are positive parameters.

d F (x; θ, β), dx

x > 0,

(1.1.1023)

(1.1.1024)

G. G. Hamedani

226

β

Remark 1.354. Taking G (x) = 1 − e−x , x ≥ 0, the cd f (1.1.1023) can be expressed as (   )#θ G (x) 2 , x ≥ 0, F (x; θ, β) = 1 − exp − 1 − G (x) "

which is a special case of the cd f mentioned in Remark 1.2.

1.1.511

Beta Exponential Pareto (BEP)

The cd f and pd f of BEP are given, respectively, by β

1 F (x; α, β, λ, a, b) = B (a, b)

Z 1−e−α( λx ) 0

wa−1 (1 − w)b−1 dw, x ≥ 0,

(1.1.1025)

and d F (x; α, β, λ, a, b), x > 0, dx where α, β, λ, a, b are all positive parameters. f (x; α, β, λ, a, b) =

(1.1.1026)

x β

Remark 1.355. Taking Q (x) = 1 − e−α( λ ) , x ≥ 0, the cd f (1.1.1025) can be written as F (x; α, β, λ, a, b) =

1 B (a, b)

Z Q(x) 0

wa−1 (1 − w)b−1 dw, x ≥ 0,

which has been mentioned in Remark 1.15.

1.1.512

Gompertz Flexible Weibull (GoFW)

The cd f and pd f of GoFW are given, respectively, by  

F (x; α, γ, η, θ) = 1 − e

θ γ

 h  η i−γ 1− exp −eαx− x

,

x ≥ 0,

(1.1.1027)

x > 0,

(1.1.1028)

and d F (x; α, γ, η, θ), dx where α, γ, η, θ are all positive parameters. f (x; α, γ, η, θ) =

Remarks 1.356. (a) The cd f (1.1.1027) can be expressed as  n h  η io θ 1− exp γeαx− x γ

or as

F (x; α, γ, η, θ) = 1 − e

−

F (x; α, γ, η, θ) = 1 − e

  θ γ

,

  η   1− exp −γeαx− x     η  exp −γeαx− x



x ≥ 0,

, x ≥ 0.

Introduction 227   η Taking G (x) = 1 − exp −γeαx− x , x ≥ 0, in the last formula, the cd f (1.1.1027) can be written as −

F (x; α, γ, η, θ) = 1 − e

 h θ γ

G(x) 1−G(x)

i

, x ≥ 0,

which is a special case of the cd f mentioned in Remark 1.2. (b) The parameter θ does not play any role in defining the cd f (1.1.1027) and hence should be replace with 1.

1.1.513

Type I Half-Logistic Rayleigh (TIHLR)

The cd f and pd f of TIHLR are given, respectively, by 2

1 − e−αλx F (x; α, λ) = 2, 1 + e−αλx

x ≥ 0,

(1.1.1029)

and f (x; α, λ) =

d F (x; α, λ), dx

x > 0,

(1.1.1030)

where α, λ are positive parameters. Remark 1.357. The cd f (1.1.1029) is a special case of the one given in Remark 1.8.

1.1.514

Log-Beta Modified Weibull (LBMW)

The cd f and pd f of LBMW are given, respectively, by

F (x; λ, µ, σ, a, b) =

1 B (a, b)

Z G(x) 0

wa−1 (1 − w)b−1 dw,

x ∈ R,

(1.1.1031)

and d F (x; λ, µ, σ, a, b), x ∈ R, (1.1.1032) dx where  λ > 0,
µ ∈ R,x σ > 0, a > 0, b > 0 are parameters and G (x) = 1 − exp − exp x−µ σ exp (λe ) , x ∈ R. f (x; λ, µ, σ, a, b) =

Remark 1.358. The cd f (1.1.1031) is as the one given in Remark 1.15 in which Q (x) = G (x).

1.1.515

Transmuted Exponential-G (TE-G)

The cd f and pd f of TE-G are given, respectively, by h ih i F (x; λ, θ, η) = 1 − (1 − G (x; η))λ 1 + θ (1 − G (x; η))λ ,

x ∈ R, and

(1.1.1033)

G. G. Hamedani

228

h i f (x; λ, θ, η) = λg (x; η) (1 − G (x; η))λ−1 1 − θ + 2θ (1 − G (x; η))λ ,

(1.1.1034)

x ∈ R, where λ > 0, θ ∈ [−1, 1], are parameters and G (x; η) is a baseline cd f with pd f g (x; η) which may depend on a parameter vector η .

1.1.516

New Mixture of Exponential-Gamma (NMEG)

The cd f and pd f of NMEG are given, respectively, by   βγ (β, λx) 1 F (x; β, λ) = 1 − e−x + , 1+β Γ (β)

(1.1.1035)

x ≥ 0, and

λ f (x; β, λ) = 1+β

(

β(λx)β−1 1+ Γ (β)

)

e−x , x > 0,

(1.1.1036)

where β > 0, λ > 0 are parameters.

1.1.517

New Generalized Akash (NGA)

The cd f and pd f of NGA are given, respectively, by

and

F (x; β, θ, η) = 1−    η 2θx + (θx)2 − βθ (2 (1 + θx) − βθ) 1 +  x ≥ β, 2 (θ2 + η) f (x; β, θ, η) =

d F (x; β, θ, η), x > β, dx

(1.1.1037)

(1.1.1038)

where β > 0, θ > 0, θ2 > −η are parameters. Remark 1.359. Characterizations of the cd f (1.1.1037) are similar to those stated for the cd f (1.1.243).

1.1.518

T-R { Y } Power Series (T-R { Y } PS)

The cd f and pd f of T-R {Y} PS are given, respectively, by F (x) = 1 − and

C (θ (1 − FT (QY (FR (x))))) , x ∈ R, C (θ)

(1.1.1039)

Introduction

229

d F (x) , x ∈ R, (1.1.1040) dx where θ > 0 and R , T, Y , C (θ) , FR , FT and QY are as defined for the cd f (1.1.981). f (x) =

Remark 1.360. Characterizations of the cd f (1.1.1039) are similar to those stated for the cd f (1.1.981).

1.1.519

New Odd Log-Logistic Chen (NOLL-Ch)

The cd f and pd f of NOLL-Ch are given, respectively, by    γ β λ 1−ex 1−e F (x; β, γ, δ, λ) =    γ   , x ≥ 0, β β λ 1−ex δλ 1−ex 1−e +e

(1.1.1041)

and

  xβ β−1 xβ λ 1−e

λβx f (x; β, γ, δ, λ) =

e e

  γ−1   β β λ 1−ex (δ−1)λ 1−ex

1−e

e

   γ   2 β β λ 1−ex δλ 1−ex 1−e +e      β λ 1−ex × γ + (δ − γ) 1 − e ,

(1.1.1042)

where β, γ, δ, λ are all positive parameters.

1.1.520

Transmuted Generalized Lindley (TGL)

The cd f and pd f of TGL are given, respectively, by

x ≥ 0, and

 α   θx −θx e × F (x; α, θ, λ) = 1 − 1 + θ+1      θx −θx (1 + λ) − λ 1 − 1 + e , θ+1

(1.1.1043)

d F (x; α, θ, λ), x > 0, (1.1.1044) dx where α > 0, θ > 0, λ ∈ [−1, 1] are parameters. 
−θx α θx Remark 1.361. Taking G (x) = 1 − 1 + θ+1 e , x ≥ 0, the cd f (1.1.1043) can be written as f (x; β, γ, δ, λ) =

F (x; α, θ, λ) = (1 + λ) G (x) − λG2 (x) ,

which has appeared in Remark 1.28.

G. G. Hamedani

230

1.1.521

Beta Exponentiated Weibull Geometric (BEWG)

The cd f and pd f of BEWG are given, respectively, by

F (x; α, β, θ, p, a, b) =

1 B (a, b)

Z 1−G(x) 0

wa−1 (1 − w)b−1 dw, x ≥ 0,

(1.1.1045)

and f (x; α, β, θ, p, a, b) =

d F (x; α, β, θ, p, a, b), dx

x > 0,

where α, β, θ, a, b all positive, p ∈ (0, 1) are parameters and G (x) =

(1.1.1046)   β θ 1− 1−e−(αx)   , β θ 1−p 1−e−(αx)

x ≥ 0.

Remarks 1.362. (a) Taking Q (x) = 1 − G (x), the cd f (1.1.1045) will be the same as the one given in Remark 1.15. (b) There is a typo in (5) and (7) on page 946 of their paper.

1.1.522

Beta Exponentiated Nadarajah-Haghighi (BENH)

The cd f and pd f of BENH are given, respectively, by 1 F (x; α, θ, λ, a, b) = B (a, b)

Z {1−e1−(1+λx)α }θ 0

wa−1 (1 − w)b−1 dw,

(1.1.1047)

x ≥ 0, and d F (x; α, θ, λ, a, b), x > 0, (1.1.1048) dx where α, θ, λ, a, b are all positive parameters. n o α θ Remark 1.363. Taking Q (x) = 1 − e1−(1+λx) , x ≥ 0, the cd f (1.1.1047) will be the same as the one given in Remark 1.15. f (x; α, θ, λ, a, b) =

1.1.523

Exponentiated Additive Weibull (EAW)

The cd f and pd f of EAW are given, respectively, by

and

h iλ θ β F (x; α, β, θ, λ, µ) = 1 − e−(αx +µx ) , x ≥ 0,

d F (x; α, β, θ, λ, µ), x > 0, dx where α, β, θ, λ, µ are all positive parameters. f (x; α, β, θ, λ, µ) =

(1.1.1049)

(1.1.1050)

Remark 1.364. The cd f (1.1.1049) is a special case of the ones given in Remarks 1.11 and 1.95.

Introduction

1.1.524

231

Transmuted Kumaraswamy Lindley (TKL)

The cd f and pd f of TKL are given, respectively, by a b )    θx × F (x; θ, λ, a, b) = 1 − 1 − 1 − e−θx 1 + θ+1 (    a b ) θx 1 + λ 1 − 1 − e−θx 1 + , x ≥ 0, θ+1 (

(1.1.1051)

and d F (x; θ, λ, a, b), x > 0, (1.1.1052) dx where θ > 0, λ ∈ [−1, 1], a > 0 and b > 0 are parameters.
a  θx , x ≥ 0, the cd f (1.1.1051) will be the Remark 1.365. Taking G (x) = 1 − e−θx 1 + θ+1 same as the cd f (1.1.1033). f (x; θ, λ, a, b) =

1.1.525

Weighted Power Lindley (WPL)

The cd f and pd f of WPL are given, respectively, by        γ βc + 1 , x + 1θ γ βc + 2 , x      F (x; β, θ, c) = , Γ βc + 1 + 1θ Γ βc + 2

(1.1.1053)

x ≥ 0, and



c β +1




β βθ xβ+c−1 1 + xβ e−θx    , x > 0, f (x; β, θ, c) =   Γ βc + 1 + θ1 Γ βc + 2

(1.1.1054)

where β, θ, c are all positive parameters.

1.1.526

Marshall-Olkin Extended Power Lomax (MOEPL)

The cd f and pd f of MOEPL are given, respectively, by

x ≥ 0, and

F (x; α, β, γ, λ) = 1 − h 1+

α

,
iγ x β −α λ

h
β iγ−1 αβγλ−β xβ−1 1 + λx f (x; α, β, γ, λ) = nh o2 , x > 0,
iγ x β 1+ λ −α

where α, β, γ, λ are all positive parameters.

(1.1.1055)

(1.1.1056)

G. G. Hamedani

232

Remark 1.366. The pd f given by the authors is incorrect. The corrected version is given in (1.1.1056).

1.1.527

Unit Modified Burr-III (UMBIII)

The cd f and pd f of UMBIII are given, respectively, by "

x F (x; α, β, γ) = 1 + γ 1−x 

−β #−α/γ

,

(1.1.1057)

d F (x; α, β, γ), x > 0, dx

(1.1.1058)

0 ≤ x ≤ 1, and f (x; α, β, γ) = where α, β, γ are all positive parameters. Remarks 1.367. (a) The cd f (1.1.1057) is a special case of the cd f given in Remark 1.51 (published in (2017)) F (x; c, θ, k) =

(

K (x; θ) 1+ 1 − K (x; θ) 

−c )−k

, x ∈ R,

for K (x) = x, 0 ≤ x ≤ 1, c = β and k = αγ . (b) The statements of the propositions 2.280 and 2.281 are incorrectly copied from the original ones due to Hamedani.

1.1.528

Reflected Power Function (RPF)

The cd f and pd f of RPF are given, respectively, by F (x; β, γ, θ) = 1 −

(θ − x)γ , θ − β ≤ x ≤ θ, βγ

(1.1.1059)

and d F (x; β, γ, θ), θ − β < x < θ , dx where β, γ, are all positive parameters. f (x; β, γ, θ) =

(1.1.1060)

Remark 1.368. The cd f (1.1.1059) can be looked upon as a special case of the cd f (1.1.187).

1.1.529

Transmuted Odd Fréchet-G (TOFr-G)

The cd f and pd f of TOFr-G are given, respectively, by iθ ( h iθ ) h −

F (x; λ, θ, ζ) = e

and

G(x;ζ) G(x;ζ)

−

1 + λ − λe

G(x;ζ) G(x;ζ)

, x ∈ R,

(1.1.1061)

Introduction

233

d F (x; λ, θ, ζ), x ∈ R, (1.1.1062) dx where λ ∈ [−1, 1], θ > 0, ζ are parameters and G (x; ζ) is a baseline cd f which may depend on the parameter ζ. f (x; λ, θ, ζ) =

−

Remark 1.369. Taking K (x) = e

h

i G(x;ζ) θ G(x;ζ)

, x ∈ R, the cd f (1.1.1061) can be written as

F (x; λ, θ, ζ) = (1 + λ) K (x) − λK (x)2 , x ∈ R, which has been given in Remark 1.28.

1.1.530

Unit- Birnbaum-Saunders (UBS)

The cd f and pd f of UBS are given, respectively, by " (    )# 1 log x 1/2 β 1/2 F (x; α, β) = 1 − Φ − − − , x ≥ 0, α β logx

(1.1.1063)

and " 1/2  3/2 # 1 β β √ f (x; α, β) = − + − log x logx 2αβx 2π    1 logx β × exp + +2 , 2 2α β logx

(1.1.1064)

x > 0, where α > 0, β > 0 are parameters.

1.1.531

erf-G (erf-G)

The cd f and pd f of erf-G are given, respectively, by   G (x; θ) , F (x; θ) = erf G (x; θ)

x ∈ R,

(1.1.1065)

and

  2  G(x;θ) 2g (x; θ) − G(x;θ) f (x; θ) = √ , x ∈ R, π (1 − G (x; θ))2

(1.1.1066)

where θ ∈ Θ ⊆ R p , Θ represents parameter space, G (x; θ) is a baseline cd f with the correR 2 sponding pd f g (x; θ) and erf (z) = √2π 0z e−t dt.

G. G. Hamedani

234

1.1.532

New Unit-Lindley (NUL)

The cd f and pd f of NUL are given, respectively, by    1−x (θ + x) exp −θ , 0 ≤ x ≤ 1, F (x; θ) = x (1 + θ) x and    θ2 1−x f (x; θ) = 3 exp −θ , 0 < x < 1, x (1 + θ) x

(1.1.1067)

(1.1.1068)

where θ > 0 is a parameter.

2

θ Remark 1.370. The function h (x|θ) = (θ+x)x 2 given on page 56 of the authors paper is not the hazard function. It is the reverse hazard function.

1.1.533

Odd Generalized Exponentiated Inverse Lomax (OGE-IL)

The cd f and pd f of OGE-IL are given, respectively, by     −α a β     1 + x     F (x; α, β, a, b) = 1 − exp −b   −α   ,     1 − 1 + βx

(1.1.1069)

x ≥ 0, and

d F (x; α, β, a, b), x > 0, (1.1.1070) dx where α, β, a, b are all positive parameters.  −α Remark 1.371. Taking G (x) = 1 + βx , x ≥ 0, the cd f (1.1.1069) can be expressed as    a G (x) F (x; α, β, a, b) = 1 − exp −b , x ≥ 0, 1 − G (x) f (x; α, β, a, b) =

which is a special case of the cd f mentioned in Remark 1.2.

1.1.534

Marshall-Olkin Inverse-Lomax (MO-IL)

The cd f and pd f of MO-IL are given, respectively, by

and

 −α 1 + βx  F (x; α, β, θ) =  −α  , x ≥ 0, 1 − (1 − θ) 1 − 1 + βx f (x; α, β, θ) =

where α, β, θ are all positive parameters.

d F (x; α, β, θ), x > 0, dx

(1.1.1071)

(1.1.1072)

Introduction 235 −α  , x ≥ 0, the cd f (1.1.1071) can be written as Remark 1.372. Taking G (x) = 1 + βx F (x; α, β, θ) =

G (x) , x ≥ 0, θ + (1 − θ) G (x)

which is a special case of the cd f mentioned in Remark 1.38.

1.1.535

Odd Lindley-Rayleigh (OLR)

The cd f and pd f of OLR are given, respectively, by 2

F (x; α, θ) = 1 − and

n  2 o αeθx /2 exp −α eθx /2 − 1 , x ≥ 0, (α + 1)

f (x; α, θ) =

d F (x; α, θ), x > 0, dx

(1.1.1073)

(1.1.1074)

where α, θ are positive parameters. Remark 1.373. Taking G (x) = 1 − e−θx

2 /2

, x ≥ 0, the cd f (1.1.1073) can be expressed as    G (x) α + G (x) F (x; α, θ) = 1 − exp −α , x ≥ 0, (α + 1) G (x) G (x)

which is a special case of the cd f (1.1.261) as well that of the cd f s mentioned in Remarks 1.91 and 1.126.

1.1.536

Cubic Transmuted Gompertz (CTG)

The cd f and pd f of CTG are given, respectively, by i h − α eβx −1) × F (x; α, β, λ1, λ2 ) = 1 − e β ( h i − α eβx −1) − 2α eβx −1) 1 + (λ1 + λ2 ) e β ( − λ2 e β ( ,

(1.1.1075)

x ≥ 0, and

d F (x; α, β, λ1 , λ2 ) , x > 0, (1.1.1076) dx where α > 0, β > 0, λ1 ∈ (−1, 1), λ2 ∈ (−1, 1) and −2 ≤ λ1 + λ2 ≤ 1 are parameters. f (x; α, β, λ1 , λ2 ) =

Remarks 1.374. (a) The formula (10) on page 108 of the authors paper is incorrect. The − α eβx −1) corrected version is given in (1.1.1075). (b)Taking G (x) = 1 − e β ( , x ≥ 0, the cd f (1.1.1075) can be written as

h i F (x; α, β, λ1 , λ2 ) = G (x) 1 + λ1 + (λ2 − λ1 )G (x) − λ2 G (x)2 , x ≥ 0,

which has been mentioned in Remark 1.217.

G. G. Hamedani

236

1.1.537

Type II Topp-Leone-Power Lomax (TIITL-PL)

The cd f and pd f of TIITL-PL are given, respectively, by F (x; α, β, γ, λ) = 1 −

(

x ≥ 0, and

λ
λ + xβ

!γ "

2−

λ
λ + xβ

!γ #)α

,

(1.1.1077)

d F (x; α, β, γ, λ), x > 0, (1.1.1078) dx where α, β, γ, λ are all positive parameters.  γ λ Remark 1.375. Taking G (x) = 1 − λ+xβ , x ≥ 0, the cd f (1.1.1077) can be expressed ( ) as f (x; α, β, γ, λ) =

  α  , x ≥ 0, F (x; α, β, γ, λ) = 1 − G (x) 2 − G (x)

which is a special case of the cd f mentioned in Remark 1.93.

1.1.538

Odd Lindley Inverse Exponential (OLINEX)

The cd f and pd f of OLINEX are given, respectively, by ( " #)
α + 1 − e−θ/x e−θ/x
exp −α F (x; α, θ) = 1 − , 1 − e−θ/x (1 + α) 1 − e−θ/x

(1.1.1079)

x ≥ 0, and

f (x; α, θ) =

d F (x; α, θ), x > 0, dx

(1.1.1080)

where α > 0, θ > 0 are parameters. Remark 1.376. Taking G (x) = e−θ/x , x ≥ 0, the cd f (1.1.1079) can be written as    G (x) α + G (x) F (x; α, θ) = 1 − exp −α , x ≥ 0, (α + 1) G (x) G (x)

which is a special case of the cd f (1.1.261) as well that of the cd f s mentioned in Remarks 1.91 and 1.126.

1.1.539

New Generalized Rayleigh (NGR)

The cd f and pd f of NGR are given, respectively, by  2η  x F (x; η, θ) = 1 − exp − 2 , x ≥ 0, 2θ

and

(1.1.1081)

Introduction

f (x; η, θ) =

d F (x; η, θ), x > 0, dx

237

(1.1.1082)

where η > 0, θ > 0 are parameters. Remark 1.377. The cd f (1.1.1081) is a special case of the cd f (1.1.29) as well that of the cd f mentioned in Remark 1.11.

1.1.540

Marshall-Olkin Power Generalized Weibull (MOPGW)

The cd f and pd f of MOPGW are given, respectively, by α

β 1 − e1−(1+λx ) h i, x ≥ 0, F (x; α, β, γ, λ) = β α γ + (1 − γ) 1 − e1−(1+λx )

and

d F (x; α, β, γ, λ), x > 0, dx where α, β, γ, λ are all positive parameters. f (x; α, β, γ, λ) =

(1.1.1083)

(1.1.1084)

α

β Remark 1.378. Taking G (x) = 1 − e1−(1+λx ) , x ≥ 0, the cd f (1.1.1083) can be expressed as

F (x; α, β, γ, λ) =

G (x) , x ≥ 0, γ + (1 − γ) G (x)

which is a special case of the cd f mentioned in Remark 1.38.

1.1.541

Type II Topp-Leone Power Ishita (TIITLPI-G)

The cd f and pd f of TIITLPI-G are given, respectively, by )β    θxα (θxα + 2) 2 −2θxα F (x; α, β, θ) = 1 − 1 − 1 − 1 + e , θ3 + 2 (

(1.1.1085)

x ≥ 0, and
α 2αβθ2 θ + x2α xα−1 e−θx f (x; α, β, θ) = × θ3 + 2     1 + θxα (θxα + 2) −θxα 1− e × θ3 + 2 ( )β−1   2 θxα (θxα + 2) α 1− 1− 1+ e−2θx , x > 0, θ3 + 2 where α, β, θ are all positive parameters.

(1.1.1086)

G. G. Hamedani

238

1.1.542

Weighted Inverse Nakagami (WINK)

The cd f and pd f of WINK are given, respectively, by
m Γ m − 2a , wx 2
, x ≥ 0, F (x; a, m, w) = Γ m − 2a

(1.1.1087)

x ≥ 0, and

f (x; a, m, w) =

 m m− a 2 2 a−2m−1 − m
x e wx2 , x > 0, a w Γ m− 2

where a > 0, m > a2 , w > 0 are parameters and Γ (m, x) =

1.1.543

(1.1.1088)

R ∞ m−1 −t e dt. x t

Generalized Marshall-Olkin Poisson-G (GMOP-G)

The cd f and pd f of GMOP-G are given, respectively, by
#θ α e−λG(x) − e−λ F (x; α, λ, θ) = 1 − , x ∈ R, 1 − αe−λ − αe−λG(x) "

(1.1.1089)

and f (x; α, λ, θ) =

d F (x; α, λ, θ) , x ∈ R, dx

(1.1.1090)

where α, λ, θ are all positive parameters. Remark 1.379. Taking K (x) = 1 − e

−λG(x) −e−λ

1−e−λ

, x ∈ R, the cd f (1.1.1089) can be written as



αK (x) F (x; α, λ, θ) = 1 − K (x) + αK (x)

θ

, x ∈ R,

which is a special case of the cd f proposed by Handique et al. (2019) mentioned in Remark 1.38.

1.1.544

Half-Logistic XGamma (HLXG)

The cd f and pd f of HLXG are given, respectively, by

and

2 [1 − A (θ, x)]λ  , x ≥ 0, F (x; θ, λ) =  1 + [1 − A (θ, x)]λ f (x; θ, λ) =

where θ, λ are positive parameters.

d F (x; θ, λ) , x > 0, dx

(1.1.1091)

(1.1.1092)

Introduction

239

Remark 1.380. Taking G (x) = [1 − A (θ, x)]λ , x > 0, the cd f (1.1.1091) can be expressed as F (x; θ, λ) =

G (x) , x ≥ 0, 1 − 21 G (x)

which is a special case of the cd f proposed by Pinho et al. (2016) mentioned in Remark 1.89.

1.1.545

Log-Gamma-Generated (LGG1)

The cd f and pd f of LGG1 are given, respectively, by F (x; α, β) = and f (x; α, β) =

Z x

−∞

f (u; α, β)du, x ∈ R,

α−1 βα  − log G(x) G (x)β−1 g (x) , x ∈ R, Γ (α)

(1.1.1093)

(1.1.1094)

where α > 0, β > 0 are parameters and G (x) is a baseline cd f with the corresponding pd f g (x).

1.1.546

Log-Gamma-Generated (LGG2)

The cd f and pd f of LGG2 are given, respectively, by F (x; α, β) = and f (x; α, β) =

Z x

−∞

f (u; α, β)du, x ∈ R,

βα [− log G (x)]α−1 G (x)β−1 g (x) , x ∈ R, Γ (α)

(1.1.1095)

(1.1.1096)

where α > 0, β > 0 are parameters and G (x) is a baseline cd f with the corresponding pd f g (x). Remark 1.381. The characterizations given in Section 2.1 for the LGG1 can be stated for the LGG2 .

1.1.547

McDonald Gumbel (MG)

The cd f and pd f of MG (WLOG, for µ = 0 and σ = 1) are given, respectively, by F (x; a, b, c) = and

1 B (a, b)

Z exp[−ce−x ] 0

ua−1 (1 − u)b−1 du, x ∈ R,

(1.1.1097)

G. G. Hamedani

240

f (x; a, b, c) =


 
b−1 c exp −x − ace−x 1 − exp −ce−x , x ∈ R, B (a, b)

(1.1.1098)

where a > 0, b > 0, c > 0 are parameters.

1.1.548

Alpha-Beta Skew Logistic G (ABSLG)

The cd f and pd f of ABSLG are given, respectively, by F (x; α, β) =

Z x

and f (x; α, β) =

h

−∞

f (u; α, β)du, x ∈ R,

1 − αx − βx3


2

i + 1 e−x

K (1 + e−x )2

(1.1.1099)

, x ∈ R,

(1.1.1100)

2 2

14 2 6 4 where α > 0, β > 0 are parameters and K = 2 + α 3π + 31 21 β π + 15 αβπ is the normalizing constant.

1.1.549

Generalized Transmuted Poisson-G (GTPG)

The cd f and pd f of GTPG are given, respectively, by " # eθG(x;η) − 1 eθ − eθG(x;η) F (x; λ, θ, η) = 1+λ , x ∈ R, eθ − 1 eθ − 1

(1.1.1101)

and d F (x; λ, θ, η), x ∈ R, dx where θ > 0 and λ ∈ [−1, 1] are parameters and G (x; η) is a baseline cd f . f (x; λ, θ, η) =

Remark 1.382. Taking K (x) =

eθG(x;η) −1 , eθ −1

(1.1.1102)

x ∈ R, the cd f (1.1.1101) can be written as

F (x; λ, θ, η) = (1 + λ) K (x) − λK (x)2 , x ∈ R,

which appeared in Remark 1.28.

1.1.550

Generalized Marshall-Olkin Transmuted-G (GMOT-G)

The cd f and pd f of GMOT-G are given, respectively, by α [1 − G (x) {1 + λ − λG (x)}] F (x; α, λ, θ) = 1 − 1 − α [1 − G (x) {1 + λ − λG (x)}] 

and

θ

, x ∈ R,

(1.1.1103)

Introduction

241

d F (x; α, λ, θ), x ∈ R, (1.1.1104) dx where α > 0, θ > 0 and λ ∈ [−1, 1] are parameters, α = 1 − α and G (x) is a baseline cd f . f (x; α, λ, θ) =

Remark 1.383. Taking K (x) = (1 + λ) G (x) − λG (x)2 , x ∈ R, the cd f (1.1.1103) can be expressed as 

αK (x) F (x; αλ, θ) = 1 − 1 − αK (x)

θ

, x ∈ R,

which is a special case of the cd f proposed by Handique et al. (2019) mentioned in Remark 1.38.

1.1.551

Generalized Odd Linear Exponential (GOLE)

The cd f and pd f of GOLE are given, respectively, by aG (x; φ)c b c+ 1 − G (x; φ) 2

"

F (x; a, b, c, φ) = 1 − exp −



G (x; φ)c 1 − G (x; φ)c

2 !#

, x ∈ R, (1.1.1105)

and

f (x; a, b, c, φ) =

"

cg (x; φ)G (x; φ)c−1 (a + (b − a) G (x; φ)c )

"

exp −

#

× 3 (1 − G (x; φ)c )  2 !# aG (x; φ)c b G (x; φ)c + , x ∈ R, 1 − G (x; φ)c 2 1 − G (x; φ)c

(1.1.1106)

where a > 0, b > 0, c > 0, a + b > 0 are parameters and G (x; φ) is a baseline cd f with the corresponding pd f g (x; φ).

1.1.552

Exponentiated Odd Chen-G (EOCh-G)

The cd f and pd f of EOCh-G are given, respectively, by "

"



F (x; α, β, θ, κ) = 1 − exp −α e x ∈ R, and

β G(x;κ) 1−G(x;κ)

−1

#!#θ

,

(1.1.1107)

G. G. Hamedani

242

f (x; α, β, θ, κ) =



αβθg (x; κ)G (x; κ)β−1 e

β G(x;κ) 1−G(x;κ)

(1 − G (x; κ))β+1 "  #! 

exp −α e "

G(x;κ) 1−G(x;κ)

"



1 − exp −α e

×

β

−1

β G(x;κ) 1−G(x;κ)

×

−1

#!#θ−1

,

(1.1.1108)

x ∈ R, where α, β, θ are all positive parameters and G (x; κ) is a baseline cd f with the corresponding pd f g (x; κ).

1.1.553

Transmuted Complementary Exponential Power (TCEP)

The cd f and pd f of TCEP are given, respectively, by   θ x β ( ) α F (x; α, β, θ, λ) = (1 + λ) 1 − exp 1 − e 



x − λ 1 − exp 1 − e( α )

β

2θ

, x ≥ 0,

(1.1.1109)

and d F (x; α, β, θ, λ), x > 0, (1.1.1110) dx where α > 0, β > 0, θ > 0 and λ ∈ [−1, 1] are parameters.   θ x β ) ( Remark 1.384. Taking G (x) = 1 − exp 1 − e α , x ≥ 0, the cd f (1.1.1109) can be f (x; α, β, θ, λ) =

written as

F (x; α, β, θ, λ) = (1 + λ) G (x) − λG (x)2 , x ≥ 0,

which was discussed in Remark 1.28.

1.1.554

Hyperbolic Cosine Weibull (HCW)

The cd f and pd f of HCW are given, respectively, by F (x; α, β, λ) = and

   2eα −λxβ sinh α 1 − e , x ≥ 0, e2α − 1

   2αeαβλxβ−1 −λxβ −λxβ e cosh α 1 − e , e2α − 1 x > 0, where α > 0, β > 0, λ > 0 are parameters. f (x; α, β, λ) =

(1.1.1111)

(1.1.1112)

Introduction

1.1.555

243

Burr X Exponential-G (BXE-G)

The cd f and pd f of BXE-G are given, respectively, by (  2 )#θ FE−G (x; β, λ) F (x; β, λ, θ) = 1 − exp − , F E−G (x; β, λ) "

(1.1.1113)

x ∈ R, and f (x; β, λ, θ) =

d F (x; β, λ, θ), x ∈ R, dx

(1.1.1114) n  o G(x;β) where β > 0, λ > 0, θ > 0 are parameters FE−G (x; β, λ) = 1 − exp −λ G(x;β) ,x∈R and G (x; β) is a baseline cd f . Remark 1.385. The cd f (1.1.1113) is a special case of the cd f mentioned in Remark 1.2.

1.1.556

Extended Generalized Inverse Exponential (EGIEx)

The cd f and pd f of EGIEx are given, respectively, by F (x; α, θ, ϕ) =

(







1− 1− 1− 1−e

x ≥ 0, and f (x; α, θ, ϕ) =

β x

λ α 2

)θ

,

d F (x; α, θ, ϕ), x > 0, dx

(1.1.1115)

(1.1.1116)

where α, θ, ϕ are all positive parameters.     β λ x Remark 1.386. Taking G (x) = 1 − 1 − e , x ≥ 0, the cd f (1.1.1115) can be ex-

pressed as

n  2 oθ F (x; α, θ, ϕ) = 1 − 1 − G (x)α , x ≥ 0,

which is a special case of the cd f mentioned in Remark 1.2.

1.1.557

Weibull-Negative Binomial (WNB)

The cd f and pd f of WNB are given, respectively, by F (x; α, β, p, k) = 1 − (1 − p)k e−k(xα)

β

x ≥ 0, and f (x; α, β, p, k) = βαβ kxβ−1 (1 − p)k e−k(xα)

  β −k 1 − pe−(xα) , β

  β −(k+1) 1 − pe−(xα) ,

x > 0, where α > 0, β > 0, p ∈ (0, 1), k ∈ N+ = {1, 2, ...} are parameters.

(1.1.1117)

(1.1.1118)

G. G. Hamedani

244

1.1.558

Unit Generalized Half Normal (UGHN)

The cd f and pd f of UGHN (WLOG, for β = 1) are given, respectively, by  √  F (x; α, 1) = 1 − erf [− log (x)] / 2 ,

(1.1.1119)

0 ≤ x ≤ 1, and

r   2α 1 2 α f (x; α, 1) = − e− 2 (− log(x)) , π x log (x) 0 < x < 1, where α > 0 is a parameter and erf (u) =

1.1.559

R u −t 2 √2 e dt π 0

(1.1.1120)

is the error function.

Generalization of Exponential and Lindley (GEL)

The cd f and pd f of GEL are given, respectively, by (α + β + αβx) e−αx F (x; α, β, a, b) = 1 − 1 − 1 − α+β x ≥ 0, and





a b

,

(1.1.1121)

d F (x; α, β, a, b), x > 0, (1.1.1122) dx where α, β, a, b are all positive parameters. h ia (α+β+αβx)e−αx Remark 1.387. Taking G (x) = 1 − , x ≥ 0, the cd f (1.1.1121) can be α+β written as f (x; α, β, a, b) =

F (x; α, β, a, b) = 1 − [1 − G (x)]b , x ≥ 0,

which is a special case of the cd f mentioned in Remark 1.12 as well as the one mentioned in Remark 1.41.

1.1.560

Unit Nadarajah-Haghighi Generated (UNH-G)

The cd f and pd f of UNH-G are given, respectively, by
1 − exp 1 − (1 + G (x; ψ))α F (x; α, ψ) = , x ∈ R, 1 − exp (1 − 2α )

(1.1.1123)

and

d F (x; α, ψ), x ∈ R, dx where α > 0 is a parameter and G (x; ψ) is a baseline cd f . f (x; α, ψ) =

(1.1.1124)

Introduction Remark 1.388. Taking K (x; ψ) = can be expressed as

1−(1+G(x;ψ))α , 1−exp(1−2α)

F (x; α, ψ) = which is mentioned in Remark 1.40.

1.1.561

245

x ∈ R and λ = 2α − 1, the cd f (1.1.1123)

1 − e−λK(x;ψ) , x ∈ R, 1 − e−λ

Lomax D function Generalized Weibull (LDGW)

The cd f and pd f of LDGW are given, respectively, by 

and

θ −1/λ

e−θ+θ(1−λx ) F (x; θ, β, λ) = 1 − 1 + eθ − 1



f (x; θ, β, λ) = βθ2 1 + 

−θ+θ(1−λxθ )

e

θ e−θ+θ(1−λx )



−1/λ

eθ − 1 −1/λ

−β −1

where θ > 0, β > 0, λ ≤ 0 are parameters.

1.1.562

−β−1

−1

xθ−1 1 − λxθ eθ − 1

, x ≥ 0,

(1.1.1125)

×


− 1 −1  λ  , x > 0,

(1.1.1126)

Right Truncated Power Lomax (RTPL)

The cd f and pd f of RTPL are given, respectively, by

and


−α 1 − 1 + xβ F (x; α, β) = , x ≥ 0, 1 − 2−α f (x; α, β) =

d F (x; α, β), x > 0, dx

(1.1.1127)

(1.1.1128)

where α > 0, β > 0 are parameters. Remark 1.389. Characterizations of the cd f (1.1.1127) are similar to those of the cd f (1.1.957).

G. G. Hamedani

246

1.1.563

Exponentiated Garima (EG)

The cd f and pd f of EG are given, respectively, by    α θx −θx e F (x; α, θ) = 1 − 1 + , x ≥ 0, θ+2 and f (x; α, θ) =

d F (x; α, θ), x > 0, dx

(1.1.1129)

(1.1.1130)

where α > 0, θ > 0 are parameters. Remark 1.390. The cd f (1.1.1130) is a special case of the cd f (1.1.657) as well as a special case of the cd f (1.1.1121).

1.1.564

New Extended Weibull (NEW)

The cd f and pd f of NE-W are given, respectively, by F (x; α, θ, γ) = 1 − and

1 − 1 − e−γx

α


2

1 − (1 − θ) (1 − e−γx )2 α

!θ

, x ≥ 0,

(1.1.1131)

d F (x; α, θ), x > 0, (1.1.1132) dx where α > 0, θ > 0, γ > 0 are all positive parameters. α
2 Remark 1.391. Taking G (x) = 1 − e−γx , x ≥ 0, the cd f (1.1.1131) can be written as f (x; α, θ) =

θ G(x) F (x; α, θ, γ) = 1 − , x ≥ 0, 1 − (1 − θ) G (x) which is a special case of the cd f discussed in Remark 1.38. 

1.1.565

Kumarswamy Sushila (KwS)

The cd f and pd f of KwS are given, respectively, by "

θ

(α (θ + 1) + αθx) e− α x F (x; α, θ, a, b) = 1 − 1 − 1 − α (θ + 1)

#a !b

x ≥ 0,

(1.1.1133)

and d F (x; α, θ, a, b), x > 0, dx where α, θ, a, b are all positive parameters. f (x; α, θ, a, b) =

Remark 1.392. The cd f (1.1.1133) is the same as the cd f (1.1.1121).

(1.1.1134)

Introduction

1.1.566

247

Alpha Power Transformed Inverse Lomax (APTIL)

The cd f and pd f of APTIL (for α 6= 1) are given, respectively, by b −a

and

α(1+ x ) − 1 , F (x; α, a, b) = α−1

x ≥ 0,

(1.1.1135)

d F (x; α, a, b), x > 0, (1.1.1136) dx where α > 0 (α 6= 1), a > 0, b > 0 are parameters.
−a Remark 1.393. Taking G (x) = 1 + bx , x ≥ 0, the cd f (1.1.1135) can be written as f (x; α, a, b) =

αG(x) − 1 , x ≥ 0, α−1 which is a special case of the cd f (1.1.63). See also Remark 1.34. F (x; α, a, b) =

1.1.567

New Extended-F (NE-F)

The cd f and pd f of NE-F are given, respectively, by F (x; θ, η) = G (x; η) eθG(x;η) ,

x ∈ R,

(1.1.1137)

and  f (x; θ, η) = g (x; η) eθG(x;η) 1 − θG (x; η) , x ∈ R,

(1.1.1138)

where θ > 0, η > 0 are parameters and G (x; η) is a baseline cd f with the corresponding pd f g (x; η).

1.1.568

Topp-Leone Rayleigh (TLR)

The cd f and pd f of TLR are given, respectively, by     2 α 2 α F (x; α, η) = 1 − e−ηx 1 + e−ηx ,

x ≥ 0,

(1.1.1139)

and

f (x; α, η) =

d F (x; α, η), x > 0, dx

(1.1.1140)

where α > 0, η > 0 are parameters. Remark 1.394. The cd f (1.1.1139) is a special case of the cd f mentioned in Remark 1.42.

G. G. Hamedani

248

1.1.569

Marshall-Olkin Topp Leone-G (MOTL-G)

The cd f and pd f of MOTL-G are given, respectively, by n oα 1 − [1 − G (x; η)]2 n oα , F (x; α, c, η) = c + c 1 − [1 − G (x; η)]2

x ≥ 0,

(1.1.1141)

and

d F (x; α, c, η), x > 0, (1.1.1142) dx where α > 0, c > 0 are parameters and G (x; η) is a baseline cd f which may depend on the parameter η. f (x; α, c, η) =

Remark 1.395. Dias et al. (2016) introduced the following distribution F (x; α, λ, p) =

(


λ ) α 1 − G (x) ,
λ 1 − p G(x)

x ∈ R,

where α > 0, λ > 0 , p < 1 are parameters and G (x) is a baseline cd f . Taking λ = 2, p = 1 − c , for c > 0, in F (x; α, λ, p), we arrive at the cd f (1.1.1141). So, the proposed distribution by Khaleel et al. (2020) is a special case of the cd f F (x; α, λ, p) has been characterized in Hamedani and Safavimanesh (2017).

1.1.570

Right Truncated-X (RT-X)

The cd f and pd f of RT-X are given, respectively, by F (x; a) = FT {aG (x)} /FT (a) ,

x ∈ R,

(1.1.1143)

and f (x; a) =

a g (x) f T {aG (x)} , x ∈ R, FT (a)

(1.1.1144)

where a > 0 is a parameter, G (x) is a baseline cd f with the corresponding pd f g (x) and FT (x) , f T (x) are the cd f and pd f of a nonnegative random variable T . Remark 1.396. We characterize the RT-X distribution in the following section. Similar characterizations can be stated for the Left-Truncate-X (LT-X) distribution of Alzaatreh et al. (2020).

1.1.571

Generalized Marshall-Olkin Inverse Lindley (GMOIL)

The cd f and pd f of GMOIL are given, respectively, by F (x; α, θ, b) = 1 −

(


−θ/x
)b θ 1 α 1 − 1 + 1+θ x e

, θ 1 −θ/x 1 − α 1 − 1 + 1+θ x e

(1.1.1145)

Introduction

249

x ≥ 0, and f (x; α, θ, b) =

d F (x; α, θ, b), x > 0, dx

(1.1.1146)

where α, θ, b are all positive parameters.
−θ/x
θ 1 Remark 1.397. Taking G (x) = 1 − 1 − 1 + 1+θ , x ≥ 0, the cd f (1.1.1145) can x e be expressed as F (x; α, θ, b) = 1 −



αG(x) 1 − αG (x)

b

, x ≥ 0,

which is a special case of the cd f mentioned in Remark 1.38.

1.1.572

Generalized Reciprocal Exponential (GRE)

The cd f and pd f of GRE are given, respectively, by c3

F (x; c1 , c2 , c3 ) =

e−c1 c2

c3 x

and

e−c1 c2 x   , x ≥ 0, c3 c1 + 1 − e−c2 x

d F (x; c1 , c2 , c3 ), x > 0, dx where c1 , c2 , c3 are all positive parameters. f (x; c1 , c2 , c3 ) =

(1.1.1147)

(1.1.1148)

c3

Remark 1.398. Taking G (x) = e−c2 x , x ≥ 0, the cd f (1.1.1147) can be written as F (x; c1 , c2 , c3 ) =

(G (x))c1
c , x ≥ 0, (G (x))c1 + G (x) 1

which is a special case of the cd f mentioned in Remark 1.39.

1.1.573

Generalized Transmuted Moment Exponential (GTME)

The cd f and pd f of GTME are given, respectively, by

x ≥ 0, and

h ia F (x; β, λ, a, b) = 1 − (1 + βx) e−βx ×  h ib  −βx 1 + λ − λ 1 − (1 + βx) e ,

d F (x; β, λ, a, b), x > 0, dx where β > 0, λ ∈ [−1, 1], a > 0, b > 0 are parameters. f (x; β, λ, a, b) =

(1.1.1149)

(1.1.1150)

G. G. Hamedani

250

Remark 1.399. Taking G (x) = 1 − (1 + βx) e−βx , x ≥ 0, the cd f (1.1.1149) can be expressed as n o F (x; β, λ, a, b) = (G (x))a 1 + λ − λ (G (x))b , x ≥ 0,

which has been introduced by Nofal et al. (2019) and characterized in the current work.

1.1.574

Generalized Weighted Exponential (GWEx)

The cd f and pd f of GWEx are given, respectively, by h ib F (x; α, λ, a, b) = 1 − e−a(1+λ)αx , x ≥ 0,

and

d F (x; α, λ, a, b), x > 0, dx where α, λ, a, b are all positive parameters. f (x; α, λ, a, b) =

(1.1.1151)

(1.1.1152)

Remark 1.400. The cd f (1.1.1151) is a special case of the cd f (1.1.25).

1.1.575

Exponentiated Transmuted Length-Biased Exponential (ETLBE)

The cd f and pd f of ETLBE are given, respectively, by         x −x/γ α x −x/γ α e 1+β 1+ e , x ≥ 0, F (x; α, β, γ) = 1 − 1 + γ γ

(1.1.1153)

and d F (x; α, β, γ), x > 0, (1.1.1154) dx where α > 0, β ∈ [−1, 1] and γ > 0 are parameters.   Remark 1.401. Taking G (x) = 1 − 1 + xγ e−x/γ , x ≥ 0, the cd f (1.1.1153) can be written as h iα F (x; α, β, γ) = (1 + β) G (x) − βG (x)2 , x ≥ 0, f (x; α, β, γ) =

which is a well-known distribution. Please see the cd f (1.1.1149) as well.

Introduction

1.1.576

251

Gamma Inverse Weibull (GIW)

The cd f and pd f of GIW are given, respectively, by

ba F (x; a, b, δ) = 1 − Γ (a) ba = 1− Γ (a)

Z

−δ 1−e−x −δ e−x

0

Z ex−δ −1 0

t a−1 e−bt dt t a−1 e−bt dt,

(1.1.1155)

x ≥ 0, and f (x; a, b, δ) =

δba x−(δ+1) ex Γ (a)

−δ

 −δ  h  −δ i ex − 1 exp −b ex − 1 , x > 0,

(1.1.1156)

where a, b, δ are all positive parameters. −δ Taking K (x) = e−x , x ≥ 0, the cd f (1.1.1155) can be written as ba F (x; a, b, δ) = 1 − Γ (a)

Z

0

K(x) K(x)

t a−1 e−bt dt, x ≥ 0,

which is a generalization of the cd f (1.1.1155) and similar characterizations can be stated for the general case as well.

1.1.577

Transmuted Alpha Power-G (TAPO-G)

The cd f and pd f of TAPO-G are given, respectively, by

x ∈ R, and

  αG(x) − 1 αλ  G(x)−1 F (x; α, λ) = 1+ 1−α , α−1 α−1

(1.1.1157)

d F (x; α, λ), x ∈ R, (1.1.1158) dx where α ∈ R+ − {1} and λ ∈ [−1, 1] are parameters and G (x) is a baseline cd f . f (x; α, λ) =

Remark 1.402. Taking K (x) =

αG(x) −1 α−1 ,

x ∈ R, the cd f (1.1.1157) can be expressed as

F (x; α, λ) = (1 + λ) K (x) − λK (x)2 , x ∈ R,

which is a special case of the cd f mentioned in Remark 1.399.

G. G. Hamedani

252

1.1.578

Exponentiated Poisson-Exponential (EPE)

The cd f and pd f of EPE are given, respectively, by F (x; α, λ, θ) =

−λx

e−θe − e−θ 1 − e−θ

!α

,

x ≥ 0,

(1.1.1159)

and f (x; α, λ, θ) =

d F (x; α, λ, θ), x > 0, dx

(1.1.1160)

where α, λ, θ are all positive parameters. Remark 1.403. Taking γ = eθ , the cd f (1.1.1159) can be written as !α −λx γ1−e − 1 , x ≥ 0, F (x; α, λ, θ) = γ−1 which was mentioned in Remark 1.34.

1.1.579

Exponentiated Power Function (EPF)

The cd f and pd f of EPF are given, respectively, by

and

 β   g−x α F (x; α, β, m, g) = 1 − , g−m

m ≤ x ≤ g,

d F (x; α, β, m, g), m < x < g, dx where α > 0, β > 0, m < g are parameters. f (x; α, β, m, g) =

(1.1.1161)

(1.1.1162)

Remark 1.404. The cd f (1.1.1161) is a special case of the cd f (1.1.187).

1.1.580

Alpha Logarithmic Transformed Weibull (ALTW)

The cd f and pd f of ALTW, for α 6= 1, are given, respectively, by h  i β log α − (α − 1) 1 − e−λx F (x; α, β, λ) = 1 − , x ≥ 0, log(α)

(1.1.1163)

and d F (x; α, β, λ), x > 0, dx where α > 0 (α 6= 1), λ > 0 are parameters. f (x; α, β, λ) =

(1.1.1164)

Introduction

253

β

Remark 1.405. Taking G (x) = 1 − e−λx , x ≥ 0, the cd f (1.1.1163) can be expressed as F (x; α, β, λ) = 1 −

log[α − (α − 1) G (x)] , x ≥ 0, log(α)

which was discussed in Remark 1.35.

1.1.581

Odd Exponentiated Half-Logistic Exponential (OEHLEx)

The cd f and pd f of OEHLEx are given, respectively, by   α  −θ[1−e−λx] 1 − exp e−λx      F (x; α, θ, λ) =  , x ≥ 0, −λx −θ[1−e ]  1 + exp e−λx

(1.1.1165)

and

d F (x; α, θ, λ), x > 0, dx where α > 0, θ > 0, λ > 0 are parameters. f (x; α, θ, λ) =

(1.1.1166)

Remark 1.406. Taking G (x) = 1 − e−λx , x ≥ 0, the cd f (1.1.1165) can be written as n o α  −θG(x) 1 − exp G(x) n o  , x ≥ 0, F (x; α, θ, λ) =  1 + exp −θG(x) G(x)

which has been characterized in Afify et al. (2017).

1.1.582

Odd Burr III Exponential (OBIIIE)

The cd f and pd f of OBIIIE are given, respectively, by

and

  −αx c −k e F (x; α, c, k) = 1 + , x ≥ 0, 1 − e−αx

d F (x; α, c, k), x > 0, dx where α > 0, c > 0, k > 0 are parameters. f (x; α, c, k) =

(1.1.1167)

(1.1.1168)

Remark 1.407. Taking G (x) = 1 − e−αx , x ≥ 0, the cd f (1.1.1167) can be expressed as    −k G (x) c F (x; α, c, k) = 1 + , x ≥ 0, G (x)

which was discussed in Remark 1.51.

G. G. Hamedani

254

1.1.583

Rayleigh-Geometric (RG)

The cd f and pd f of RG are given, respectively, by n 2o x 1 − exp − 2σ 2 n o , x ≥ 0, F (x; σ, p) = x2 1 − p exp − 2σ 2

(1.1.1169)

and

f (x; σ, p) =

d F (x; σ, p) , x > 0, dx

(1.1.1170)

where σ > 0, p ∈ (0, 1) are parameters. Remark 1.408. The cd f (1.1.1169) is a special case of the cd f discussed in Remark 1.8.

1.1.584

Weibull Exponentiated Exponential (WEE)

The cd f and pd f of WEE are given, respectively, by   h  a iθ  −bx F (x; a, b, θ, φ) = 1 − exp −φ − ln 1 − 1 − e ,

(1.1.1171)

x ≥ 0, and

f (x; a, b, θ, φ) =

d F (x; a, b, θ, φ), x > 0, dx

(1.1.1172)

where a, b, θ, φ are parameters. Remark 1.409. Ghosh and Nadarajah (2018), proposed the following distribution   1 c F (x; γ, c) = 1 − exp − c [− log(1 − G (x))] , x ∈ R, γ
a where G (x) is a baseline cd f . Taking G (x) = 1 − e−bx , x ≥ 0 in F (x; γ, c), we arrive at the cd f (1.1.165). Thus, WEE is a special case of F (x; γ, c).

1.1.585

Logistic Exponential (LE)

The cd f and pd f of LE are given, respectively, by

and

h  α i−1 F (x; α, λ) = 1 − 1 + eλx − 1 , x ≥ 0, f (x; α, λ) =

where α > 0, λ > 0 are parameters.

d F (x; α, λ), x > 0, dx

(1.1.1173)

(1.1.1174)

Introduction

255

Remark 1.410. Alizadeh et al. (2017) proposed the following distribution F (x) =

G (x; λ)α , x ∈ R, G (x; λ)α + βG (x; λ)α

where G (x; λ) is a baseline cd f , which has been characterized in Hamedani (2019). Taking G (x; λ) = 1 − e−λx , x ≥ 0 in the cd f (1.1.1173) we arrive at F (x; α, λ) =

G (x; λ)α , x ≥ 0, G (x; λ)α + G (x; λ)α

which is a special case of Alizadeh et al.’s proposed distribution.

1.1.586

Generalized Gamma-G (GG-G)

The cd f and pd f of GG-G are given, respectively, by 1 F (x; α, β, ζ) = Γ (β)

Z

0



G(x;ζ) G(x;ζ)

α

wβ−1 e−w dw, x ∈ R,

(1.1.1175)

and d F (x; α, β, ζ) , x ∈ R, (1.1.1176) dx where α > 0, β > 0 are parameters and G (x; ζ) is a baseline cd f which may depend on the parameter ζ. f (x; α, β, ζ) =

Remark 1.411. Cordeiro et al. (2016) proposed the following distribution F (x; α, β, τ) =

βα Γ (α)

Z

G(x;τ) G(x;τ)

0

t α−1 e−βt dt, x ∈ R,

where G (x; τ) is a baseline cd f . The characterizations of the cd f (1.1.1175) are similar to those of the cd f F (x; α, β, τ) which has been characterized in Hamedani (2018b).

1.1.587

Exponentiated Shanker (E-Sh)

The cd f and pd f of E-Sh are given, respectively, by  α θx + θ2 + 1 −θx F (x; α, θ) = 1 − e θ2 + 1     θx = 1− 1+ 2 e−θx , x ≥ 0, θ +1

(1.1.1177)

and f (x; α, θ) = where α > 0, θ > 0 are parameters.

d F (x; α, θ), x > 0, dx

(1.1.1178)

G. G. Hamedani

256

Remark 1.412. Ghitany et al. (2013) introduced the following distribution   β α x exp [−βxα ] , x ≥ 0, F (x; α, β) = 1 − 1 + β+1

which has been characterized in Hamedani and Maadooliat (2017). Similar characterizations can be stated for the cd f (1.1.1177).

1.1.588

Extended Generalized Lindley (EGL)

The cd f and pd f of EGL are given, respectively, by h   iα λx 1 − 1 + 1+λ e−λx F (x; α, γ, λ) = h  iα h   iγ x ≥ 0,  λx λx e−λx + 1 − 1 − 1 + 1+λ e−λx , 1 − 1 + 1+λ

(1.1.1179)

and

f (x; α, γ, λ) =

d F (x; α, γ, λ), x > 0, dx

(1.1.1180)

where α, γ, λ are all positive parameters. Remark 1.413. Alizadeh et al. (2017) considered the following distribution F (x; α, β, λ)   i β α λ β 1 − 1 + 1+λ x e−λx =h  iα h  iα , x ≥ 0.  λ β λ β x e−λxβ + 1 + 1+λ x e−λxβ 1 − 1 + 1+λ h

As far as the characterizations of cd f (1.1.1179) are concerned, they will be similar to those of F (x; α, β, λ), which has been characterized in Hamedani (2017). Please see Remark 1.116 as well.

1.1.589

Ristic-Balakrishnan Odd Log-Logistic-G (RBOLL-G)

The cd f and pd f of RBOLL-G are given, respectively, by h

G(x;τ)α

i

− log G(x;τ)α +G(x;τ)α 1 F (x; α, β, τ) = 1 − t β−1 e−t dt , Γ (β) 0    1 G (x; τ)α = 1− γ β, − log , x ∈ R, Γ (β) G (x; τ)α + G (x; τ)α

Z

(1.1.1181)

and d F (x; α, β, τ), x ∈ R, (1.1.1182) dx where α > 0, β > 0 are parameters and G (x; τ) is a baseline cd f which may depend on the parameter vector τ . f (x; α, β, τ) =

Introduction

257

Remark 1.414. Cordeiro et al. (2016) proposed the following distribution )# " ( 1 Gα (x; τ) , x ∈ R, F (x; α, β, τ) = γ β, − log 1 − α Γ (β) Gα (x; τ) + G (x; τ) where α, β are positive parameters and G (x; τ) is the baseline cd f which may depend on the parameter τ . The cd f F (x; α, β, τ) has been characterized in Hamedani (2016). Similar charcterizations can be stated for the cd f (1.1.1181).

1.1.590

Burr XII Exponentiated Exponential (BrXIIEE)

The cd f and pd f of BrXIIEE are given, respectively, by ( "
a #α )−β 1 − e−bx F (x; α, β, a, b) = 1 − 1 + , x ≥ 0, a 1 − (1 − e−bx )

(1.1.1183)

and d F (x; α, β, a, b), x > 0, dx where α, β, a, b are all positive parameters. f (x; α, β, a, b) =

(1.1.1184)

Remark 1.415. Alizadeh et al. (2019) proposed the following distribution G (x; η) F (x; α, β, η) = 1 − 1 + β G (x; η) 



α − β1

,

x ∈ R,

which has been characterized in Hamedani (2019). Taking G (x; η) = 1 − e−bx the cd f (1.1.1183) will be similar to the cd f F (x; α, β, η).

1.1.591


a

, x ≥ 0,

Mixture Pareto Log-Gamma (MPLG)

The cd f and pd f of MPLG are given, respectively, by     θλ log xx0   x −θ F (x; θ, λ, x0 ) = 1 − 1 + , x ≥ x0 ,  θ + λ  x0

(1.1.1185)

and

d F (x; θ, λ, x0 ), x > x0 , dx where θ, λ, x0 are all positive parameters.   Remark 1.416. Letting Y = log xX0 , the cd f of Y can be written as   θλy F (y; θ, λ) = 1 − 1 + e−θy , y ≥ 0, θ+λ f (x; θ, λ, x0 ) =

which is a special case of the cd f discussed in Remark 1.22.

(1.1.1186)

G. G. Hamedani

258

1.1.592

Exponentiated Two Parameter Pranav (ETPP)

The cd f and pd f of ETPP are given, respectively, by )β ( "
# θx 3θx + θ2 x2 + 6 −θx e , F (x; α, β, θ) = 1 − 1 + (αθ4 + 6)

(1.1.1187)

x ≥ 0, and f (x; α, β, θ) =

d F (x; α, β, θ), x > 0, dx

(1.1.1188)

where α, β, θ are all positive parameters. Remark 1.417. Characterizations of the cd f (1.1.1187) are similar to those of the cd f s (1.1.243), (1.1.535) and (1.1.1121).

1.1.593

New Lifetime Exponential-Weibull (NLTE-W)

The cd f and pd f of NLTE-W are given, respectively, by   1 − G (x; ζ) F (x; θ, ζ) = 1 − eθG(x;ζ)

= 1 − G (x; ζ) e−θG(x;ζ) ,

(1.1.1189)

x ∈ R, and d F (x; θ, ζ) , x ∈ R, (1.1.1190) dx where θ > 0 is a positive parameter and G (x; ζ) is a baseline cd f which may depend on the parameter ζ. f (x; θ, ζ) =

Remark 1.418. Characterizations of the cd f (1.1.1189) are similar to those of the cd f (1.1.45).

1.1.594

New Cubic Rank Transmutation (NCRT)

The cd f and pd f of NCRT are given, respectively, by F (x; λ1 , λ2 ) = x + λ1 x (1 − x) + λ2 x2 (1 − x) , 0 ≤ x ≤ 1,

(1.1.1191)

and d F (x; λ1 , λ2 ) , 0 < x < 1, dx where λ1 ∈ [−1, 1], λ2 ∈ [−1, 1] and −2 ≤ λ1 + λ2 ≤ 1 are parameters. f (x; λ1 , λ2 ) =

(1.1.1192)

Remark 1.419. Taking G (x) = x, 0 ≤ x ≤ 1, the cd f (1.1.1191) can be expressed as F (x; λ1 , λ2 ) = (1 + λ1 ) G (x) + (λ2 − λ1 ) G (x)2 − λ2 G (x)3 , 0 ≤ x ≤ 1,

which was discussed in Remark 1.217.

Introduction

1.1.595

259

Alpha Power Exponentiated Exponential (APExE)

The cd f and pd f of APExE are given, respectively, by −ax c

α(1−e ) − 1 , x ≥ 0, F (x; α, a, c) = α−1

and

d F (x; α, a, c), x > 0, dx where α > 0 (α 6= 1), a > 0, c > 0 are parameters. f (x; α, a, c) =

(1.1.1193)

(1.1.1194)

c

Remark 1.420. Taking G (x) = (1 − e−ax ) , x ≥ 0, the cd f (1.1.1193) can be written as F (x; α, a, c) = which was discussed in Remark 1.34.

1.1.596

αG(x) − 1 , x ≥ 0, α−1

Odd Fréchet Inverse Lomax (OFIL)

The cd f and pd f of OFIL are given, respectively, by −

F (x; α, β, θ) = e

α i h 1+ βx −1

and

, x ≥ 0,

"     # h α i αβθ β αθ−1 β −α − 1+ βx −1 f (x; α, β, θ) = 2 1 + 1− 1+ e , x x x

(1.1.1195)

(1.1.1196)

x > 0, where α, β, θ are all positive parameters.

1.1.597

Topp-Leone Power Lindley (TLPL)

The cd f and pd f of TLPL are given, respectively, by (   2 )λ βxα α F (x; α, β, λ) = 1 − 1 + e−βx , x ≥ 0, 1+β

(1.1.1197)

and f (x; α, β, λ) =

d F (x; α, β, λ), dx

x > 0,

(1.1.1198)

where α, β, λ are all positive parameters. Remark 1.421. Ghitany et al. (2013) introduced the following distribution   βxα α F (x; α, β) = 1 − 1 + e−βx , x ≥ 0, 1+β which has been characterized in Hamedani and Maadooliat (2017). Similar characterizations can be stated for the cd f (1.1.1197).

G. G. Hamedani

260

1.1.598

Gull Alpha Power (GAP)

The cd f and pd f of GAP are given, respectively, by F (x; α) =

αG (x) , αG(x)

x ∈ R,

(1.1.1199)

and d F (x; α), x ∈ R, dx where α > 0, α 6= 1 is a parameter and G (x) is a baseline cd f . f (x; α) =

Remark 1.422. The cd f (1.1.157) is given by ( G(x;η) F (x; β, η) =

G(x;η)β β

G(x;η)

, β > 0, β 6= 1

, β=1

(1.1.1200)

, x ∈ R.

Taking β = α1 , the cd f F (x; β, η), for β 6= 1 will reduce to the cd f (1.1.1199).

1.1.599

Modi Generator (MG)

The cd f and pd f of MG are given, respectively, by
1 + αβ G (x) F (x; α, β) = αβ + G (x) G (x) = , 1 1 − 1+αβ (1 − G (x))

x ∈ R,

(1.1.1201)

and d F (x; α, β), x ∈ R, dx where α > 0, β > 0 are parameter and G (x) is a baseline cd f . f (x; α, β) =

Remark 1.423. The cd f (1.1.1201) is not new. Please see Remark 1.23 for θ =

1.1.600

(1.1.1202)

1 . 1+αβ

Cubic Transmuted Exponentiated Pareto-1 (CTEP-1-G)

The cd f and pd f of CTEP-1-G are given, respectively, by  
1 + (λ1 + λ2 − 2) ka e−ax + a −ax F (x; a, k, λ1 , λ2 ) = 1 − k e , (1 − λ2 ) k2a e−2ax

(1.1.1203)

x ≥ lnk, and

 3 − λ2 − λ1 + f (x; a, k, λ1 , λ2 ) = aka e−ax  2 (2λ2 + λ1 − 3) ka e−ax+  , 3 (1 − λ2 ) k2a e−2ax 

x. lnk , where a > 0, k > 0, λ1 ∈ [0, 1], λ2 ∈ [−1, 1] are parameters.

(1.1.1204)

Introduction

1.1.601

261

Cubic Transmuted Exponentiated Pareto-1 (CTEP-1-R)

The cd f and pd f of CTEP-1-R are given, respectively, by  
1 + (ω1 + ω2 ) ka e−ax − F (x; a, k, ω1 , ω2 ) = 1 − ka e−ax , ω2 k2a e−2ax

(1.1.1205)

x ≥ lnk, and

d F (x; a, k, ω1 , ω2 ) , (1.1.1206) dx x > lnk , where a > 0, k > 0, ω1 ∈ [−1, 1], λ2 ∈ [−1, 1] , −2 ≤ ω1 + ω2 ≤ 1 are parameters. f (x; a, k, ω1 , ω2 ) =

Remark 1.424. Taking G (x) = 1 − ka e−ax, x ≥ lnk , the cd f (1.1.1205) can be expressed as F (x; a, k, ω1, ω2 ) = (1 + ω1 ) G (x) + (ω2 − ω1 )G (x)2 − ω2 G (x)3 , x ≥ lnk, which was discussed in Remark 1.217.

1.1.602

Slashed Quasi-Gamma (SQG)

The cd f and pd f of SQG are given, respectively, by F (x; β, θ, p) =

Z x 0

f (u; β, θ, p)du, x ≥ 0,

(1.1.1207)

and !    θ pβ p x−(p+1) p 1 x p 1
Γ f (x; β, θ, p) = + F , + ,1 , 1 θ 10 β θ 10 Γ 10

(1.1.1208)

x > 0 , where β, θ, p are all positive parameters.

1.1.603

Log-Epsilon-Skew Normal (LESN)

The cd f and pd f of LESN, WLOG for θ = 0 and σ = 1, are given, respectively, by  logx  (1 − ε)Φ( ) , 0 0, τ > 0 are parameters.  ν log(x)−µ Remark 1.426. Taking G (x) = Φ , x ≥ 0, the cd f (1.1.1211) can be expressed σ as f (x; µ, σ, ν, τ) =

F (x; µ, σ, ν, τ) =

G (x)τ , x ≥ 0, G (x)τ + G (x)τ

which was discussed in Remark 1.45.

1.1.605

Slash Power Maxwell (SPM)

The cd f and pd f of SPM are given, respectively, by     2x−p 2β p + 3β 2β 3 F (x; α, β, p) = G αx ; , 1 − √ p G αx ; , 1 x ≥ 0, 2 2β πα 2β

(1.1.1213)

and     2pΓ p+3β p + 3β 2β f (x; α, β, p) = √ p x−(p+1)G αx2β ; ,1 , 2β πα 2β

(1.1.1214)

x > 0 , where α, β, p are all positive parameters and G (x; a, b) is the cd f of a gamma distribution.

1.1.606

Modified Slashed Half-Normal (MSHN)

The cd f and pd f of MSHN are given, respectively, by F (x; σ, p) =

Z x 0

f (u; σ, p) du,

x ≥ 0,

(1.1.1215)

Introduction

263

and 2p f (x; σ, p) = √ x−(p+1) N σ 2π



 p+1 2 p 1 , , , , 2 x p 2 2σ2

x > 0 , where σ, p are positive parameters and N (a, b, c, g) =

1.1.607

(1.1.1216)

R ∞ a−1 exp (−bxc − gx) dx. 0 x

One Parameter Polynomial Exponential-G (OPPE-G)

The cd f and pd f of OPPE-G are given, respectively, by    s Γ k + 1, λ G(x;η) G(x;η) , F (x; λ, η) = 1 − p (λ) ∑ ak  k+1 λ k=0

(1.1.1217)

x ∈ R , and

f (x; λ, η) = p (λ)

(

s

∑ ak

k=0



G (x; η) G (x; η)

where λ > 0 is a parameters, p (λ) =

k )

∑sk=0 ak

g (x; η) −λ  2 e G (x; η)

1

h

Γ(k+1) λk+1



G(x;η) G(x;η)

i

,

x ∈ R,

(1.1.1218)

and G (x; η) is a baseline cd f with the

corresponding pd f (x; η), which may depend on the parameter η.

1.1.608

Beta Burr Type X (BBX)

The cd f and pd f of BBX are given, respectively, by 1 F (x; ν, ω, η, υ) = B (ν, ω)

Z 1−e−(ηx)2 0

h

iυ

t ν−1 (1 − t)ω−1 dt,

(1.1.1219)

x ≥ 0 , and d F (x; ν, ω, η, υ), x > 0, (1.1.1220) dx where ν, ω, η, υ are all positive parameters. h i 2 υ Remark 1.427. Taking Q (x) = 1 − e−(ηx) , x ≥ 0, the cd f (1.1.1219) can be expressed as f (x; ν, ω, η, υ) =

1 F (x; ν, ω, η, υ) = B (ν, ω) which was discussed in Remark 1.15.

Z Q(x) 0

t ν−1 (1 − t)ω−1 dt, x ≥ 0,

G. G. Hamedani

264

1.1.609

Gamma Burr Type X (GBX)

The cd f and pd f of GBX are given, respectively, by 1 F (x; ν, η, υ) = 1 − Γ (ν)

Z −υ log 1−e−(ηx)2 h

0

i

t ν−1 e−t dt, x ≥ 0,

(1.1.1221)

and f (x; ν, η, υ) =

d F (x; ν, η, υ), x > 0, dx

(1.1.1222)

where ν, η, υ are all positive parameters. h i 2 υ Remark 1.428. Taking G (x) = 1 − e−(ηx) , x ≥ 0, the cd f (1.1.1221) can be written as F (x; ν, η, υ) = 1 −

1 Γ (ν)

Z − log G(x) 0

t ν−1 e−t dt, x ≥ 0.

In view of the Remarks 1.37 and 1.258, similar characterizations can be stated for the cd f (1.1.1221).

1.1.610

Weibull Burr Type X (WBX)

The cd f and pd f of WBX are given, respectively, by    h i 2 υω   −(ηx)   1−e   F (x; ν, ω, η, υ) = 1 − exp −ν   h i ω  , x ≥ 0, 2 υ     1 − 1 − e−(ηx)

(1.1.1223)

and

d F (x; ν, ω, η, υ), x > 0, (1.1.1224) dx where ν, ω, η, υ are all positive parameters. h i 2 υ Remark 1.429. Taking G (x) = 1 − e−(ηx) , x ≥ 0, the cd f (1.1.1223) can be expressed as     G (x) ω F (x; ν, η, υ) = 1 − exp −ν , x ≥ 0, G (x) f (x; ν, ω, η, υ) =

which is a special case of the cd f discussed in Remark 1.2.

Introduction

1.1.611

265

Extended Odd Weibull-G (ExOW-G)

The cd f and pd f of ExOW-G are given, respectively, by

and

n h iα o−1/β F (x; α, β, λ) = 1 − 1 + β eλx − 1 , f (x; α, β, λ) =

x ≥ 0,

d F (x; α, β, λ), x > 0, dx

(1.1.1225)

(1.1.1226)

where α, β, λ are all positive parameters. Remark 1.430. Taking G (x) = 1 − e−λx , x ≥ 0, the cd f (1.1.1225) can be written as G (x) F (x; α, β, λ) = 1 − 1 + β G (x) 

which was mentioned in Remark 1.163.

1.1.612



α −1/β

, x ≥ 0,

Sinh Inverted Exponential (SIE)

The cd f and pd f of SIE are given, respectively, by F (x; δ, θ) =

2eδ eδ − 1

and

    −θ/x −1 ,
2 cosh δe

f (x; δ, θ) =

x ≥ 0,

d F (x; δ, θ), x > 0, dx

(1.1.1227)

(1.1.1228)

where δ, θ are positive parameters. Remarks 1.431. (a) Kharazmi and Saadatinik (2018) introduced the following distribution F (x; δ) =

2eδ eδ − 1


2 (cosh(δG (x)) − 1) , x ∈ R,

where δ > 0 is a parameter and G (x) is a baseline cd f . Taking G (x) = e−θ/x, x ≥ 0, in F (x; δ), we arrive at the cd f (1.1.1227). The cd f F (x; δ) has been characterized in Hamedani (2019). (b) Ahmad (2019) proposed the following distribution (cd f (1.1.265)) F (x; α, σ) =

2eα (eα − 1)

     −σx2 cosh α 1 − e − 1 , x ≥ 0, 2

where α, σ are positive parameters, which in a way similar to (1.1.1227).

G. G. Hamedani

266

1.1.613

Exponential Skew-Normal (ESN)

The cd f and pd f of ESN are given, respectively, by 2 σ2 F (x; µ, σ, τ, δ) = e 2τ2 A (x) , τ

x ∈ R,

(1.1.1229)

and   2 − x−µ + σ2 x − µ σ δσ f (x; µ, σ, τ, δ) = e ( σ ) 2τ2 ΦB − , ; −δ , x ∈ R, τ σ τ τ where √1 2π 1−δ2

1.1.614

(1.1.1230)

µ ∈ R, σ > 0, τ > 0, |δ| ≤ 1 are parameters, ΦB (h, k; ρ) =   t−µ R − 1 2 (u2 +v2 −2ρuv) x ( σ ) Φ t−µ − σ , δσ ; −δ dt. 2(1−δ ) e dudv and A (x) = e B −∞ −∞ −∞ σ τ τ

Rk Rh

T-Burr (T-Burr{Y })

The cd f and pd f of T-Burr{Y } are given, respectively, by n h io F (x; c, k) = FT QY 1 − (1 + xc )−k , x ≥ 0,

(1.1.1231)

and

f (x; c, k) = ckxc−1 (1 + xc )−k−1

n h io fT QY 1 − (1 + xc )−k n h io , fY QY 1 − (1 + xc )−k

x ≥ 0,

(1.1.1232)

where c, k are positive parameters, FT is the cd f of a random variable T , QY (·) is quantile function of a random variable Y . Remark 1.432. Characterizations similar to those of cd f (1.1.981) can be stated for the cd f (1.1.1231).

1.1.615

Transmuted General (T-G)

The cd f and pd f of T-G are given, respectively, by F (x; λ) = G (x) + λ

G (x) G(x) , x ∈ R, 1 + G (x)

(1.1.1233)

and "

f (x; λ) = 1 − λ +

2λ (1 + G (x))2

#

g (x) , x ∈ R,

(1.1.1234)

where λ ∈ [−1, 1] is a parameter and G (x) is a baseline cd f with the corresponding pd f g (x).

Introduction

1.1.616

267

Odd Gamma Weibull-Geometric (OGWG)

The cd f and pd f of OGWG are given, respectively, by ! c e(βx) − 1 1 γ α, , F (x; α, β, c, p) = Γ (α) 1− p

x ≥ 0,

(1.1.1235)

and  α−1 c c cβc xc−1 e(βx) e(βx) − 1 × Γ (α)(1 − p) ! c 1 − e(βx) exp , 1− p

f (x; α, β, c, p) =

x > 0, where α > 0, β > 0, c > 0, p ∈ [0, 1) are parameters and γ (α, z) =

1.1.617

(1.1.1236)

R z α−1 −t e dt. 0t

Generalized Inverted Kumarswamy Generated (GIKw-G)

The cd f and pd f of GIKw-G are given, respectively, by

and

 α β F (x; α, β, γ, η) = 1 − (1 − Gγ (x; η)) ,

x ∈ R,

(1.1.1237)

d F (x; α, β, γ, η), x ∈ R, (1.1.1238) dx where α > 0, β > 0, γ > 0 are parameters and G (x; η) is a baseline cd f which may depend on the parameter η. f (x; α, β, γ, η) =

Remark 1.433. Taking K (x; η) = Gγ (x; η), the cd f (1.1.1237) becomes  β F (x; α, β, γ, η) = 1 − (1 − K (x; η))α ,

x ∈ R.

Mahmoud et al. (2015), proposed the following distribution n h
β ia ob F (x; α, β, a, b, η) = 1 − 1 − 1 − 1 − (K (x; η))α , x ∈ R,

which was discussed in Remark 1.12. It is clear that the cd f (1.1.1237) is a special case of the cd f F (x; α, β, a, b, η).

G. G. Hamedani

268

1.1.618

Gamma Kumarswamy-G (GKw-G)

The cd f and pd f of GKw-G are given, respectively, by F (x; α, a, b) =

1 Γ (α)

Z {1−G(x)a }−b −1 0

t α−1 e−t dt, x ∈ R,

(1.1.1239)

and d F (x; α, a, b), x ∈ R, dx where α > 0, a > 0, b > 0 are parameters and G (x) is a baseline cd f . f (x; α, a, b) =

(1.1.1240)

b

Remark 1.434. Taking K (x) = 1 − {1 − G (x)a } , x ∈ R, the cd f (1.1.1239) can be expressed as 1 F (x; α, a, b) = Γ (α)

Z

K(x) K(x)

0

t α−1 e−t dt,

x ∈ R.

Cordeiro et al. (2016) proposed the following distribution F (x; α, β, τ) =

βα Γ (α)

Z

0

K(x;τ) K(x;τ)

t α−1 e−βt dt, x ∈ R,

which reduces to the cd f (1.1.1239), for β = 1. Please see Remark 1.37.

1.1.619

Weibull Burr XII (WBXII)

The cd f and pd f of WBXII are given, respectively, by h i F (x; α, β, c, k) = 1 − exp −α {k log(1 + xc )}β , x ≥ 0,

(1.1.1241)

and

d F (x; α, β, c, k), x > 0, dx where α, β, c, k are all positive parameters. f (x; α, β, c, k) =

(1.1.1242)

Remark 1.435. Taking G (x) = 1 − (1 + xc )−k , x ≥ 0, the cd f (1.1.1241) can be written as h i F (x; α, β, c, k) = 1 − exp −α {− log (1 − G (x))}β , x > 0. Ghosh and Nadarajah (2018), proposed the following distribution   1 c F (x; γ, c) = 1 − exp − c [− log(1 − G (x))] , x ∈ R, γ

where G (x) is a baseline cd f . The cd f (1.1.1241) is a special case of the cd f F (x; γ, c) mentioned in Remark 1.60.

Introduction

1.1.620

269

Odd Generalized Gamma-G (OGG-G or GG-G)

The cd f and pd f of OGG-G are given, respectively, by 1 F (x; α, β, η) = Γ (β)

Z

0



 G(x;η) α G(x;η)

t β−1 e−t dt, x ∈ R,

(1.1.1243)

and d F (x; α, β, η), x ∈ R, (1.1.1244) dx where α, β are positive parameters and G (x; η) is a baseline cd f which may depend on the parameter η. f (x; α, β, η) =

Remark 1.436. Cordeiro et al. (2016) proposed the following distribution βα F (x; α, β, τ) = Γ (α)

Z

0

K(x;τ) K(x;τ)

t α−1 e−βt dt, x ∈ R,

which was mentioned in Remark 1.37. The characterizations of the cd f (1.1.1243) are similar to those of the cd f F (x; α, β, τ).

1.1.621

Marshall-Olkin Odd Burr III-G (MOOB-G)

The cd f and pd f of MOOB-G are given, respectively, by  −c−b  G(x;η) 1 + G(x;η) ( ) , x ∈ R, F (x; α, c, b, η) =   −c −b G(x;η) 1 − (1 − α) 1 − 1 + G(x;η)

(1.1.1245)

and d F (x; α, b, c, η), x ∈ R, (1.1.1246) dx where α > 0, b > 0, c > 0 are parameters and G (x; η) is a baseline cd f which may depend on the parameter η. h  c i−b Remark 1.437. Taking K (x) = 1 + G(x;η) , x ∈ R , the cd f (1.1.1245) can be exG(x;η) pressed as f (x; α, b, c, η) =

F (x; α, c, b, η) =

K (x) , x ∈ R, α + (1 − α) K (x)

which is a special case of the cd f (1.1.417).

G. G. Hamedani

270

1.1.622

Gamma Power Half-Logistic (GPHL)

The cd f and pd f of GPHL are given, respectively, by 1 F (x; α, β, δ) = Γ (δ)



Z − log

2 β 1+eαx

0



t δ−1 e−t dt, x ≥ 0,

(1.1.1247)

and f (x; α, β, δ) =

d F (x; α, β, δ), x > 0, dx

(1.1.1248)

where α, β, δ are all positive parameters. Remark 1.438. Taking G (x) = 1 − F (x; α, β, δ) =

2 β, 1+eαx

1 Γ (δ)

x ≥ 0 , the cd f (1.1.1247) can be written as

Z − log(1−G(x)) 0

t δ−1 e−t dt, x ≥ 0,

which is a special case of the cd f mentioned in Remark 1.24. Please see Remark 1.258 as well.

1.1.623

Topp-Leone Weibull-Lomax (TLWLx)

The cd f and pd f of TLWLx are given, respectively, by "

(

1 − (1 + bx)−a F (x; α, θ, a, b) = 1 − exp −2 (1 + bx)−a 

α )#θ

, x ≥ 0,

(1.1.1249)

and d F (x; α, θ, a, b), x > 0, dx where α, θ, a, b are all positive parameters. f (x; α, θ, a, b) =

(1.1.1250)

Remark 1.439. Taking G (x) = 1 − (1 + bx)−a , x ≥ 0 , the cd f (1.1.1249) can be expressed as     θ G (x) α F (x; α, θ, a, b) = 1 − exp −2 , x ≥ 0, G (x)

which is a special case of the cd f mentioned in Remark 1.2.

Introduction

1.1.624

271

Minimum Weibull-Burr (minWB)

The cd f and pd f of minWB are given, respectively, by b

F (x; a, b, c, k) = 1 − (1 + xc )−k e−ax , x ≥ 0,

(1.1.1251)

and d F (x; a, b, c, k), x > 0, dx where a, b, c, k are all positive parameters. f (x; a, b, c, k) =

(1.1.1252)

Remark 1.440. The cd f (1.1.1251) is a special case of the cd f (1.1.95) for θ = a1 = 1 and ai = 0, i = 2, ..., n.

1.1.625

Box-Cox Gamma-G (BCG-G)

The cd f and pd f of BCG-G are given, respectively, by 1 F (x; δ, λ) = 1 − Γ (δ)

Z

G(x;φ)−λ −1 λ

0

t δ−1 e−t dt, x ∈ R,

(1.1.1253)

and d F (x; δ, λ) , x ∈ R, (1.1.1254) dx where δ, λ are positive parameters and G (x; φ) is a baseline cd f which may depend on the parameter φ. f (x; δ, λ) =

Remark 1.441. Taking K (x) = G (x; φ)−λ , x ∈ R, the cd f (1.1.1253) can be written as 1 F (x; δ, λ) = 1 − δ λ Γ (δ)

Z

K(x) K(x)

0

uδ−1 e−u/λ du, x ∈ R,

which is the same as the cd f (1.1.1155).

1.1.626

Modified Beta Generalized Linear Failure Rate (MBGLFR)

The cd f and pd f of MBGLFR are given, respectively, by F (x; α, λ, θ, a, b) =   α 2 − λx+ θ2 x2   c1−e  

1 B (a, b) and

Z

0

  α 2 − λx+ θ2 x2   1−(1−c) 1−e  

t a−1 (1 − t)b−1 dt , x ≥ 0,

(1.1.1255)

G. G. Hamedani

272

d F (x; α, λ, θ, a, b), dx x > 0, where α, λ, θ, a, b are all positive parameters. f (x; α, λ, θ, a, b) =

"

c 1−e

Remark 1.442. Taking Q (x) =

  #α 2 − λx+ θ2 x2

"

1−(1−c) 1−e

expressed as

1 F (x; α, λ, θ, a, b) = B (a, b)

  #α 2 − λx+ θ2 x2

Z Q(x) 0

(1.1.1256)

, x ≥ 0, the cd f (1.1.1255) can be

t a−1 (1 − t)b−1 dt, x ≥ 0,

which was mentioned in Remark 1.15.

1.1.627

New Modified Burr III (NMBIII)

The cd f and pd f of NMBIII are given, respectively, by

and

 −k F(x; c, k, λ) = 1 + x−c e−λx , x ≥ 0, f (x; c, k, λ) =

d F (x; c, kλ), x > 0, dx

(1.1.1257)

(1.1.1258)

where c, k, λ are all positive parameters. Remark 1.443. Taking K (x) = 1 + x−c e−λx as


−1

, x ≥ 0, the cd f (1.1.1257) can be written

1 − K (x) F (x; c, k, λ) = 1 + K (x) 



−k

, x ≥ 0,

which is a special case of the cd f mentioned in Remark 1.51.

1.1.628

Inverted Modified Lindley (IML)

The cd f and pd f of IML are given, respectively, by   θx−1 −θ/x −θ/x F(x; θ) = 1 + e e , x ≥ 0, 1+θ

(1.1.1259)

and

i θx−2 −2θ/x h θ/x −1 f (x; θ) = e (1 + θ) e + 2θx − 1 , 1+θ x > 0 , where θ > 0 is a parameter.

(1.1.1260)

Introduction

1.1.629

273

Type II General Inverse Exponential (TIIGIE)

The cd f and pd f of TIIGIE are given, respectively, by ) ( [1 − G (x)]θ , x ∈ R, F(x; λ, θ) = exp −λ 1 − [1 − G (x)]θ

(1.1.1261)

and

d F(x; λ, θ), x ∈ R, dx where λ > 0, θ > 0 are parameters and G (x) is a baseline cd f . f (x; θ) =

(1.1.1262)

Remark 1.444. The cd f (1.1.1261) can be expressed as (  θ ) G(x) F(x; λ, θ) = exp −λ  θ , x ∈ R. 1 − G (x)

Alizadeh et al. (2016) proposed the following distribution (  θ ) K (x; γ) F (x; α, γ) = exp − , x ∈ R, 1 − K (x; γ)

where K (x; θ) is a baseline cd f . Characterizations similar to those of F (x; α, γ) can be stated for the cd f (1.1.1261). Please see Remark 1.52 for additional information.

1.1.630

Exponentiated Half-Logistic Lomax (EHLLx)

The cd f and pd f of EHLLx are given, respectively, by ( )α 1 − (1 + bx)−θ , x ≥ 0, F(x; α, θ, b) = 1 + (1 + bx)−θ

(1.1.1263)

and f (x; α, θ, b) =

d F(x; α, θ, b), x > 0, dx

(1.1.1264)

where α, θ, b are all positive parameters. Remark 1.445. Taking G (x) = 1 − (1 + bx)−θ , x ≥ 0, the cd f (1.1.1263) can be written as  α G (x) F(x; λ, θ) = , x ≥ 0. 2 − G (x) Dias et al. (2016) introduced the following distribution F (x; α, λ, p) =

(


λ ) α 1 − G (x) ,
λ 1 − p G(x)

x ∈ R,

G. G. Hamedani

274

where α > 0, λ > 0 , p < 1 are parameters and G (x) is a baseline cd f . Taking λ = 1, p = −1, the cd f F (x; α, λ, p) can be expressed as α  G (x) , x ∈ R, F (x; α, λ, p) = 2 − G (x)

which for x ≥ 0, will be the cd f (1.1.1263).

1.1.631

Generalized Gamma-Generalized Inverse Weibull (GG-GIW)

The cd f and pd f of GG-GIW are given, respectively, by

x ≥ 0, and

1 F(x; α, β, d, p) = 1 −   Γ dp

Z

0



θ α

( αx )

 β p

d

t p −1 e−t dt,

d F(x; α, β, d, p), x > 0, dx where α, β, d, p are all positive parameters. f (x; α, β, d, p) =

Remark 1.446. Taking G (x) =

1 , 1+θαβ−1 x−β

(1.1.1265)

(1.1.1266)

x ≥ 0, the cd f (1.1.1265) can be expressed as

1 F(x; α, β, d, p) = 1 −   Γ dp

Z

0



G(x) G(x)

p

d

t p −1 e−t dt, x ≥ 0,

which was discussed in Remark 1.441.

1.1.632

Log-Weighted Exponential (log-WE)

The cd f and pd f of log-WE are given, respectively, by F(x; α, λ) = and

i 1 λh x (α + 1) − xαλ , 0 ≤ x ≤ 1, α

 λ (α + 1) λ−1  x 1 − xαλ , 0 < x < 1, α where α > 0, λ > 0 are parameters. f (x; α, λ) =

1.1.633

(1.1.1267)

(1.1.1268)

Generalized Raised Cosine (GENRC)

The cd f and pd f of GENRC, WLOG for µ = 0, σ = 1, are given, respectively, by   1 λ F(x; α, λ) = 1+x+ sin(πx) , (1.1.1269) 2 2π −1 ≤ x ≤ 1, and

Introduction

275

f (x; α, λ) = C (α + λ cos (πx)), − 1 < x < 1,

(1.1.1270)

where |λ| ≤ |α| < ∞ are parameters and C is the normalizing constant.

1.1.634

Sine Kumarswamy-G (SK-G)

The cd f and pd f of SK-G are given, respectively, by π  b F(x; a, b) = cos [1 − G (x)a ] , x ∈ R, 2 and

f (x; a, b) =

π  abπg (x) b−1 b G (x)a−1 [1 − G (x)a ] sin [1 − G (x)a ] , 2 2

(1.1.1271)

(1.1.1272)

x ∈ R, where a > 0, b > 0 are parameters and G (x) is a baseline cd f with the corresponding pd f g (x).

1.1.635

Extended Exp-G (EE-G)

The cd f and pd f of EE-G are given, respectively, by F(x; α, β, η) = and

G (x; η)α G (x; η)α + 1 − G (x; η)β

, x ∈ R,

d F(x; α, β, η), x ∈ R, dx where α > 0, β > 0 are parameters and G(x; η) is a baseline cd f . f (x; a, b) =

(1.1.1273)

(1.1.1274)

Remark 1.447. Cordeiro et al. (2016) introduced the following distribution F (x; α, θ, τ) =

(G (x; τ))αθ h iα , x ∈ R. (G (x; τ))αθ + 1 − (G (x; τ))θ

Taking α = 1 in F (x; α, θ, τ), we have F (x; 1, θ, τ) =

(G (x; τ))θ (G (x; τ))θ + 1 − (G (x; τ))θ

, x ∈ R.

The characterizations of the cd f (1.1.1273) will be similar to those of the cd f F (x; 1, θ, τ) which have appeared in Hamedani (2019).

G. G. Hamedani

276

1.1.636

Weighted Exponential (WE)

The cd f and pd f of WE, WLOG for α = µ = 1, are given, respectively, by 2  3 F(x) = 1 − e− 2 x , x ≥ 0,

and

  3 3 f (x) = 3e− 2 x 1 − e− 2 x ,

1.1.637

(1.1.1275)

x > 0.

(1.1.1276)

Lindley Negative-Binomial (LNB)

The cd f and pd f of LNB are given, respectively, by

and

pGL (x; θ) F(x; k, p, θ) = 1 − (1 − p) GL (x; θ) 

f (x; k, p, θ) =

k

, x ≥ 0,

(1.1.1277)

d F (x; k, p, θ), x > 0, dx

where k > 0, p ∈ (0, 1), θ > 0 are parameters and GL (x; θ) = 1 − the Lindley distribution.

(1.1.1278) 1+θ+θx 1+θ




e−θx , x ≥ 0, is

Remark 1.448. Hadique et al. (2019) proposed the following distribution F (x; α, a, b, η) =

(



αG(x; η) 1− 1 − αG(x; η)

a )b

, x ∈ R,

where G (x; η) is a baseline cd f . Taking G (x; η) = GL (x; θ) , x ≥ 0, a = 1, α = 1/p and b = k, we arrive at the cd f (1.1.1277). The cd f F (x; α, β, η) has been characterized in Hamedani (2019).

1.1.638

Marshall-Olkin Transmuted-G (MOT-G)

The cd f and pd f of MOT-G are given, respectively, by F(x; λ, θ, η) = and

G (x; η) [1 + λ − λG (x; η)] , x ∈ R, θ + (1 − θ) G (x; η) [1 + λ − λG (x; η)]

(1.1.1279)

d F(x; λ, θ, η), x ∈ R, (1.1.1280) dx where |λ| ≤ 1, θ > 0 are parameters and G (x; η) is a baseline distribution which may depend on the parameter vector η. f (x; λ, θ, η) =

Introduction

277

Remark 1.449. Taking K (x) = G (x; η) [1 + λ − λG (x; η)] , x ∈ R, the cd f (1.1.1279) can be expressed as F(x; λ, θ, η) =

K (x) , x ∈ R, θ + (1 − θ) K (x)

which is a special case of the cd f mentioned in Remark 1.38.

1.1.639

Ratio of Two Independent Weibull and Lindley (RTIWL)

The cd f and pd f of RTIWL are given, respectively, by F(x; b, c) =

Z x 0

f (u; b, c)du, x ≥ 0,

(1.1.1281)

and

f (x; b, c) =

2xc2 2 b (c + 1)

Z ∞ 0

 2 2 
x u u3 + u2 exp − 2 − cu du, x > 0, b

(1.1.1282)

where b > 0, c > 0 are parameters.

1.1.640

Product of Two Independent Weibull and Lindley (PTIWL)

The cd f and pd f of PTIWL are given, respectively, by F(x; b, c) =

Z x

f (u; b, c)du, x ≥ 0,

(1.1.1283)

  x u2 cx 1+ exp − 2 − du, x > 0, u b u

(1.1.1284)

0

and 2c2 f (x; b, c) = 2 b (c + 1)

Z ∞ 0

where b > 0, c > 0 are parameters. Remarks 1.450. (a) We believe there is a typo in the equation (1.1.32) of Hassan et al.’s 2 R − u c2 paper. It should be F (x; b, c) = 1 − 0∞ 1 − e b2 x2 c+1 (u + 1) e−cu du, otherwise F (0) as

given in (1.1.32) would be 1 and limx→∞ F (x; b, c) would be 0. (b) Characterizations similar to those of the cd f (1.1.1281) can be stated for the cd f (1.1.1283).

1.1.641

Modified Kies Generalized (MKi-G)

The cd f and pd f of MKi-G are given, respectively, by    G (x; η) F(x; α, η) = 1 − exp − , 1 − G (x; η)

x ∈ R, and

(1.1.1285)

278

G. G. Hamedani

d F(x; α, η), x ∈ R, (1.1.1286) dx where α > 0 is a parameter and G(x; η) is a baseline cd f which may depend on the parameter vector η. f (x; α, η) =

Remark 1.451. The cd f (1.1.1285) is a special case of the cd f mentioned in Remark 1.2.

1.1.642

Ratio Exponentiated General (RE-G)

The cd f and pd f of RE-G are given, respectively, by F(x; α, η) =

2G (x; η)β , x ∈ R, 1 + G (x; η)α

(1.1.1287)

and   2G (x; η)β−1 g (x; η) β + (β − α) G (x; η)α f (x; α, η) = , x ∈ R,  2 1 + G (x; η)α

(1.1.1288)

where α and β are two shape parameters satisfying β 6= 0 and β ≥ α2 , or β = 0 and α < 0 and G (x; η) is a baseline cd f which may depend on the parameter η.

1.1.643

Gompertz Exponential (GoEp)

The cd f and pd f of GoEp are given, respectively, by   θ γ

[1−eγλx]

, x ≥ 0,

(1.1.1289)

d F(x; θ, γ, λ), x > 0, dx

(1.1.1290)

F(x; θ, γ, λ) = 1 − e and f (x; θ, γ, λ) = where θ, γ, λ are all positive parameters.

Remark 1.452. Taking G (x) = 1 − e−γλx , x ≥ 0, the cd f (1.1.1289) can be expressed as −

F(x; θ, γ, λ) = 1 − e

 h θ γ

G(x) G(x)

i

, x ≥ 0,

which is a special case of the cd f mentioned in Remark 1.2.

Introduction

1.1.644

279

Minimum Gumbel Burr (minGuBu)

The cd f and pd f of minGuBu are given, respectively, by  −α   −a F(x; α, β, a, b) = 1 − 1 + xβ 1 − e−bx , x ≥ 0,

and

 −α−1 f (x; α, β, a, b) = 1 + xβ

(


−a ) abx−a−1 1+ xβ e−bx , x > 0, −a +αβxβ−1 1 − e−bx

(1.1.1291)

(1.1.1292)

where α, β, a, b are all positive parameters.

Remark 1.453. Taking G1 (x) and G2 (x) , x ∈ R, as two cd f s, the cd f (1.1.1291) can be written, in general, as F(x; α, β, a, b) = 1 − G1 (x) G2 (x) , x ∈ R.

1.1.645

Transmuted Power Gompertz (TPG)

The cd f and pd f of TPG are given, respectively, by    θ   θ  2 − αβ eβx −1 − αβ eβx −1 F(x; α, β, θ, λ) = (1 + λ) 1 − e −λ 1−e ,

(1.1.1293)

x ≥ 0, and d F (x; α, β, θ, λ), dx x > 0, where α > 0, β > 0, θ > 0, and λ ∈ [−1, 1] are parameters. f (x; α, β, θ, λ) =

 θ  − αβ eβx −1

Remark 1.454. Taking G (x) = 1 − e as

(1.1.1294)

, x ≥ 0, the cd f (1.1.1293) can be expressed

F(x; α, β, θ, λ) = (1 + λ) G (x) − λG (x)2 , x ≥ 0,

which was discussed in Remark 1.28.

1.1.646

Burr X-G (BX-G)

The cd f and pd f of BX-G are given, respectively, by (  2 )#θ G (x; φ) F(x; θ, φ) = 1 − exp − , 1 − G (x; φ) "

x ∈ R, and

(1.1.1295)

G. G. Hamedani

280

f (x; θ, φ) =

d F (x; θ, φ), x ∈ R, dx

(1.1.1296)

where θ > 0, φ > 0 are parameters. Remark 1.455. The cd f (1.1.1295) is a special case of the cd f mentioned in Remark 1.119.

1.1.647

Hamza (Hamza)

The cd f and pd f of Hamza are given, respectively, by F(x; α, β) =

Z x 0

f (u; α, β)du, x ≥ 0,

(1.1.1297)

and   β6 βx6 −βx α+ e , x > 0, f (x; α, β) = αβ5 + 120 6

(1.1.1298)

where α > 0, β > 0 are parameters.

1.1.648

Inverse Lomax-G (IL-G)

The cd f and pd f of IL-G are given, respectively, by 

βH (x; υ) F(x; α, β, υ) = 1 + H (x; υ) and

−α

, x ∈ R,

(1.1.1299)

d F(x; α, β, υ), x ∈ R, (1.1.1300) dx where α > 0, β > 0, υ > 0 are parameters and H (x; υ) is a baseline cd f which may depend on the parameter υ. f (x; α, β) =

Remark 1.456. The cd f (1.1.1299) can be written as  α H (x; υ) F(x; α, β, υ) = , x ∈ R, β + (1 − β) H (x; υ) which is a special case of the cd f mentioned in Remark 1.38.

1.1.649

Zubair-Inverse Lomax (ZIL)

The cd f and pd f of ZIL are given, respectively, by   −2β λ exp α 1 + x −1 F(x; α, β, λ) = , x ≥ 0, eα − 1 and

(1.1.1301)

Introduction

281

d F(x; α, β, λ), x > 0, (1.1.1302) dx where α > 0, β > 0, λ > 0 are parameters. −2β  Remark 1.457. Taking G (x) = 1 + λx , x ≥ 0, the cd f (1.1.1301) can be expressed as f (x; α, β, λ) =

eαG(x) − 1 , x ≥ 0, eα − 1 which is a special case of the cd f mentioned in Remark 1.34. F(x; α, β, υ) =

1.1.650

Skew Scale Mixtures Normal (SSMN)

The cd f and pd f of SSMN, WLOG for µ = 0, σ = 1, are given, respectively, by F(x; λ) = and

Z x

f (u; λ)du,

x ∈ R,

(1.1.1303)

f (x; λ) = 2 f 0 (x) Φ (λx) ,

x ∈ R,

(1.1.1304)

−∞

where λ R> 0 is a parameter, Φ (x) , φ (x) are the cd f and pd f of standard normal and f0 (x) = 0∞ φ (x; κ (u)) dH (u; τ), H (u; τ) is a cd f of a positive random variable U index by the parameter vector τ and κ (·) is a strictly positive function.

1.1.651

Akash (Akash)

The cd f and pd f of Akash are given, respectively, by   θx (θx + 2) −θx F(x; θ) = 1 − 1 + e , x ≥ 0, θ2 + 2

(1.1.1305)

and

f (x; θ) =

d F(x; θ), dx

x > 0,

(1.1.1306)

where θ > 0 is a parameter. Remark 1.458. The cd f (1.1.1305) is a special case of the cd f (1.1.61).

1.1.652

A Generalization of Sujatha (AGS)

The cd f and pd f of AGS are given, respectively, by   θx (αθx + θ + 2α) −θx F(x; θ, α) = 1 − 1 + e , θ2 + θ + 2α

x ≥ 0, and

(1.1.1307)

G. G. Hamedani

282

f (x; θ, α) =

d F(x; θ, α), dx

x > 0,

(1.1.1308)

where θ > 0, α > 0 are parameters. Remark 1.459. The cd f (1.1.1307) is a special case of the cd f (1.1.61).

1.1.653

Two-Parameter Sujatha (TPS)

The cd f and pd f of TPS are given, respectively, by   θx (θx + θ + α) −θx F(x; θ, α) = 1 − 1 + e , αθ2 + θ + 2 x ≥ 0, and f (x; θ, α) =

d F(x; θ, α), dx

x > 0,

(1.1.1309)

(1.1.1310)

where θ > 0, α > 0 are parameters. Remark 1.460. The cd f (1.1.1309) is almost identical to the cd f (1.1.1307).

1.1.654

New Two-Parameter Sujatha (NTPS)

The cd f and pd f of NTPS are given, respectively, by   θx (θx + αθ + 2) −θx F(x; θ, α) = 1 − 1 + e , θ2 + αθ + 2 x ≥ 0, and f (x; θ, α) =

d F(x; θ, α), dx

x > 0,

(1.1.1311)

(1.1.1312)

where θ > 0, α > 0 are parameters. Remark 1.461. The cd f (1.1.1311) is almost identical to the cd f s (1.1.1307) and (1.1.1309).

1.1.655

Another Two-Parameter Sujatha (ATPS)

The cd f and pd f of ATPS are given, respectively, by   αθx (θx + αθ + 2) −θx F(x; θ, α) = 1 − 1 + e , θ2 + αθ + 2α x ≥ 0, and f (x; θ, α) =

d F(x; θ, α), dx

x > 0,

(1.1.1313)

(1.1.1314)

where θ > 0, α > 0 are parameters. Remark 1.462. The cd f (1.1.1313) is almost identical to the cd f s (1.1.1307), (1.1.1309) and (1.1.1311).

Introduction

1.1.656

283

Lomax Inverse Weibull (LxIW)

The cd f and pd f of LxIW are given, respectively, by h
b i −β exp − ax−1 h i , F(x; β, a, b) = 1 − 1 + 1 − exp − (ax−1 )b 

x ≥ 0, and

f (x; β, a, b) =

(1.1.1315)

d F(x; β, a, b), x > 0, dx

(1.1.1316)

where β, a, b are all positive parameters. h
b i Remark 1.463. Taking G (x) = exp − ax−1 , x ≥ 0, the cd f (1.1.1315) can be expressed as  G (x) −β 1 − G (x)  −β 1 = 1+ 1 − G (x)
β = 1 + G (x) , x ≥ 0,

 F(x; β, a, b) = 1 + 1 +

which is a special case of the cd f (1.1.57).

1.1.657

Odd Burr Generalized Rayleigh (OBGR)

The cd f and pd f of OBGR are given, respectively, by

x ≥ 0, and

F(x; β, δ, θ) = 1 − 

   δθ 2 β −x 1− 1−e


2 βδ 1 − e−x +

f (x; β, δ, θ) =

h

1−


iδ 2 β 1 − e−x

θ ,

d F(x; β, δ, θ), x > 0, dx

(1.1.1317)

(1.1.1318)

where β, δ, θ are all positive parameters.   2 β Remark 1.464. Taking G (x) = 1 − e−x , x ≥ 0, the cd f (1.1.1317) can be written as F(x; β, δ, θ) = 1 − n

G (x)δθ δ

G (x) + G (x)

δ

oθ , x ≥ 0.

Brito et al. (2017) introduced the following distribution

G. G. Hamedani

284

F (x; a, b, λ) =

(

G (x; λ)a 1− 1− G (x; λ)a + G (x; λ)a 

2 )b

, x ∈ R,

which is more general than the cd f (1.1.1317) as 2 can be replaced with any parameter θ > 0.

1.1.658

Marshall-Olkin Lehmann Burr X (MOLBX)

The cd f and pd f of MOLBX are given, respectively, by h   i 2 a θ 1 − 1 − 1 − e−x F(x; a, δ, θ) = 
 , x ≥ 0, 2 a θ 1 − δ 1 − 1 − e−x

and

f (x; a, δ, θ) =

d F(x; a, δ, θ), x > 0, dx

(1.1.1319)

(1.1.1320)

where a, δ, θ are all positive parameters. h   iθ 2 a Remark 1.465. Taking G (x) = 1 − 1 − 1 − e−x , x ≥ 0, the cd f (1.1.1319) can be expressed as F(x; a, δ, θ) =

G (x) , x ≥ 0, δ + (1 − δ) G (x)

which is a special case of the cd f (1.1.417).

1.1.659

Transmuted Topp-Leone Weibull (TTL-W)

The cd f and pd f of TTL-W are given, respectively, by

and

h i h i b α b 2α F(x; a, b, α, λ) = (1 + λ) 1 − e−2(ax) − λ 1 − e−2(ax) , x ≥ 0,

(1.1.1321)

d F(x; a, b, α, λ), x > 0, (1.1.1322) dx where a > 0, b > 0, α > 0, λ ∈ [−1, 1] are parameters. h i b α Remark 1.466. Taking G (x) = 1 − e−2(ax) , x ≥ 0, the cd f (1.1.1321) can be written as f (x; a, b, α, λ) =

F(x; a, b, α, λ) = (1 + λ) G (x) − λG (x)2 , x ≥ 0,

which was mentioned in Remark 1.183.

Introduction

1.1.660

285

Burr X Generalized Burr XII (BXGBXII)

The cd f and pd f of BXGBXII are given, respectively, by F(x; a, b, c, λ) ( =

"

(  −2 )#) h i−c −b a 1 − exp −λ 1 − exp − 1 − (x + 1) −1 1 − e−λ

, x ≥ 0,

(1.1.1323)

and d F(x; a, b, c, λ), x > 0, (1.1.1324) dx where a, b, c, λ are all positive parameters. " (  −2 )# h i−c Remark 1.467. Taking G (x) = 1 − exp − 1 − (xa + 1)−b −1 , x ≥ 0, the f (x; a, b, c, λ) =

cd f (1.1.1323) can be expressed as

F(x; a, b, c, λ) = which was mentioned in Remark 1.316.

1.1.661

1 − exp {−λG (x)} , x ≥ 0, 1 − e−λ

Generalized Odd Generalized Exponential Fréchet (GOGEFr)

The cd f and pd f of GOGEFr are given, respectively, by β   −e−α( ax )b   , x ≥ 0, F(x; α, β, a, b) = 1 − exp b  −α( ax )  1−e 

and

d F(x; α, β, a, b), dx where α, β, a, b are all positive parameters. f (x; α, β, a, b) =

x > 0,

a b

(1.1.1325)

(1.1.1326)

Remark 1.468. Taking G (x) = e−α( x ) , x ≥ 0, the cd f (1.1.1325) can be written as    −G (x) β F(x; α, β, a, b) = 1 − exp , x ≥ 0, G (x)

which is a special case of the cd f mentioned in Remark 1.2.

G. G. Hamedani

286

1.1.662

Topp-Leone Lindley (TLLi)

The cd f and pd f of TLLi are given, respectively, by #β  1 + λ + λx 2 −2λx e F(x; β, λ) = 1 − 1+λ "   #β 1 + λ + λx −λx 2 = 1− ( )e , 1+λ "



(1.1.1327)

x ≥ 0, and f (x; β, λ) =

d F(x; β, λ), dx

x > 0,

(1.1.1328)

where β > 0, λ > 0 are parameters. Remark 1.469. The cd f (1.1.1327) is a special case of the cd f (1.1.657).

1.1.663

Logarithmic Transformed Inverse Weibull (LTIW)

The cd f and pd f of LTIW are given, respectively, by h i −α log 1 + δ − δexp(−λx ) F(x; δ, α, λ) = 1 − , x ≥ 0, log(δ) and d F(x; δ, α, λ), x > 0, dx where δ > 0 (δ 6= 1), α > 0, λ > 0 are parameters. f (x; δ, α, λ) =

Remark 1.470. Taking G (x) =

−α )

1−δexp(−λx 1−δ

F(x; δ, α, λ) = 1 −

(1.1.1329)

(1.1.1330)

, x ≥ 0, the cd f (1.1.1329) can be expressed as

log[δ + (1 − δ) G (x)] , x ≥ 0, log(δ)

which was mentioned in Remark 1.35.

1.1.664

Kumarswamy Alpha Power-G (KAP-G)

The cd f and pd f of KAP-G, for α 6= 1, are given, respectively, by (

and

"

αG(x) − 1 F(x; α, a, b) = 1 − 1 − α−1

#a )b

, x ∈ R,

d F(x; α, a, b), x ∈ R, dx where α (α 6= 1) , a, b are all positive parameters and G (x) is a baseline cd f . f (x; α, a, b) =

(1.1.1331)

(1.1.1332)

Introduction Remark 1.471. Taking K (x) =

αG(x) −1 α−1 ,

287

x ∈ R, the cd f (1.1.1331) can be written as b

F(x; α, a, b) = 1 − {1 − K (x)a } , x ≥ 0,

which is a special case of the cd f s mentioned in Remarks 1.21 and 1.93.

1.1.665

Flexible Weibull Burr XII (FWBXII)

The cd f and pd f of FWBXII are given, respectively, by i h β F(x; α, β, c, k) = 1 − exp −eαx− x [1 + xc ]−k , x ≥ 0,

(1.1.1333)

and



  β αx− βx c−1 c −1 f (x; α, β, c, k) = α + 2 e + ckx [1 + x ] × x h i β exp −eαx− x [1 + xc ]−k , x > 0,

(1.1.1334)

where α, β, c, k are all positive parameters.

1.1.666

Extended Poisson Lomax (EPLx)

The cd f and pd f of EPLx are given, respectively, by

F(x; α, β, λ, θ) =

(

)   n o2 θ 
 α 1 − exp −λ 1 − exp − 1 + xβ−1 1 − e−λ

, x ≥ 0, (1.1.1335)

and d F(x; α, β, λ, θ), x > 0, (1.1.1336) dx where α, β, λ, θ are all positive parameters.   n  
α o2 θ −1 Remark 1.472. Taking G (x) = 1 − exp − 1 + xβ , x ≥ 0, the cd f f (x; α, β, λ, θ) =

(1.1.1335) can be expressed as

F(x; α, β, λ, θ) = which was mentioned in Remark 1.316.

1 − e−λG(x) , x ≥ 0, 1 − e−λ

G. G. Hamedani

288

1.1.667

Unit Johnson SU (UJSU )

The cd f and pd f of UJSU , WLOG for µ − 0, σ = 1, are given, respectively, by     x −1 F(x) = Φ sinh log , 0 ≤ x ≤ 1, (1.1.1337) 1−x and

f (x) =

1 q  x (1 − x) 1 + log

x 1−x

    x −1 , φ sinh log
 1−x

0 < x < 1, where Φ (x) and φ (x) are cd f and pd f of the standard normal.

(1.1.1338)

Chapter 2

Characterizations of Distributions We present our characterizations (i) − (iv) in four subsections.

2.1 Characterizations Based on Two Truncated Moments This subsection is devoted to the characterizations of the new distributions listed in Chapter 1 based on the ratio of two truncated moments. Our first characterization employs a theorem due to Glänzel (1987), see Theorem 2.1 of Appendix 2.4 . The result, however, holds also when the interval H is not closed, since the condition of the Theorem is on the interior of H. Proposition 2.1. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =
−1 λ + β − (λ + β + αβλx) e−αλx and q2 (x) = q1 (x) e−λx for x > 0. The random variable X has pd f (1.1.2) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−λx , x > 0. 2 Proof. Suppose the random variable X has pd f (1.1.2), then (1 − F (x)) E [q1 (X) | X ≥ x] =

(1 + α)2 e−λx , x > 0, α (λ (1 + α) + αβ)

(1 − F (x))E [q2 (X) | X ≥ x] =

(1 + α)2 e−λx , x > 0. 2α (λ (1 + α) + αβ)

and

Further, q1 (x) −λx e < 0 , f or x > 0. 2 Conversely, if ξ is of the above form, then ξ (x) q1 (x) − q2 (x) = −

s0 (x) =

ξ0 (x) q1 (x) = λ, ξ (x) q1 (x) − q2 (x)

x > 0,

290

G. G. Hamedani

and consequently s (x) = λx, x > 0. Now, according to Theorem 2.1, X has density (1.1.2)



Corollary 2.1. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.1. The random variable X has pd f (1.1.2) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = λ, x > 0. ξ (x) q1 (x) − q2 (x) Proof. Let q2 (x) be as in Proposition 2.1 as well, then

or

ξ0 (x) q1 (x) ξ0 (x) = = λ, x > 0, ξ (x) q1 (x) − q2 (x) ξ (x) − e−λx ξ0 (x) − λξ (x) = −λe−λx ,

or

or

o 1 d n o d n −λx e ξ (x) = e−2λx , dx 2 dx 1 e−λx ξ (x) = e−2λx , 2

or 1 ξ (x) = e−λx , x > 0. 2  Corollary 2.2. The general solution of the differential equation in Corollary 2.1 is  Z  −1 λx −λx ξ (x) = e − λe (q1 (x)) q2 (x) dx + D ,

where D is a constant. We like to point out that one set of functions satisfying the above differential equation is given in Proposition 2.1 with D = 0. Clearly, there are other triplets (q1 , q2 , ξ) which satisfy conditions of Theorem 2.1. A Proposition, two Corollaries similar to Proposition 2.1, Corollary 2.1 and Corollary 2.2 can be stated for each of the following remaining distributions. For each of these distributions, however, we give below, the functions q1 , q2 and ξ corresponding to Theorem 2.1.

Characterizations of Distributions

291

Proposition 2.2. Let : Ω → R be a continuous random variable and let q1 (x) =  X h i−β −β+1  G(x;η) G (x; η) exp θβ G(x;η) and q2 (x) = q1 (x) [G (x; η)]−β for x ∈ R. Then, the

random variable X has pd f (1.1.6) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

o 1n 1 + [G (x; η)]−β , x ∈ R. 2

Corollary 2.3. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.2. The random variable X has pd f (1.1.6) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) −βg (x; η) [G (x; η)]−β−1 = , ξ (x) q1 (x) − q2 (x) 1 − [G (x; η)]−β

x ∈ R.

Corollary 2.4. The general solution of the differential equation in Corollary 2.3 is  n o−1 Z −β −β−1 −1 ξ (x) = 1 − [G (x; η)] βg (x; η) [G (x; η)] (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.3. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  α+1 
 1−θ λ(1−α)x
2 1 1 − e−λx − 1+α 1 − e−λ(1+α)x e and q2 (x) = q1 (x) 1 − e−λαx for α x > 0. Then, the random variable X has pd f (1.1.14) if and only if the function ξ defined in Theorem 2.1 is of the form   2  1 −λαx ξ (x) = 1+ 1−e , x > 0. 2 Corollary 2.5. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.3. The random variable X has pd f (1.1.14) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation
2αλe−λαx 1 − e−λαx ξ0 (x) q1 (x) =
2 , x > 0. ξ (x) q1 (x) − q2 (x) 1 − 1 − e−λαx Corollary 2.6. The general solution of the differential equation in Corollary 2.5 is

   2 −1  Z   −1 −λαx −λαx −λαx ξ (x) = 1 − 1 − e − 2αλe 1−e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.4. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =   β αβ −1 1 − e− x and q2 (x) = q1 (x) e− x for x > 0. Then, the random variable X has pd f (1.1.22) if and only if the function ξ defined in Theorem 2.1 is of the form

G. G. Hamedani

292

 β 1 1 + e− x , x > 0. 2 Corollary 2.7. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.4. The random variable X has pd f (1.1.22) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ (x) =

β

ξ0 (x) q1 (x) βe− x , =  ξ (x) q1 (x) − q2 (x) x2 1 − e− βx

x > 0.

Corollary 2.8. The general solution of the differential equation in Corollary 2.7 is  n o−1  Z −1 − βx −2 − βx ξ (x) = 1 − e − βx e (q1 (x)) q2 (x) dx + D , where D is a constant.

Proposition 2.5. Let X : Ω → (a, b) be a continuous random variable and let q1 (x) = 1−θ (((x−β)3 +(β−a)3 )) 3 for x ∈ (a, b). Then, the random θ and q2 (x) = q1 (x) (x − β) (1+λ)−2λ( α3 ((x−β)3 +(β−a)3 )) variable X has pd f (1.1.40) if and only if the function ξ defined in Theorem 2.1 is of the form   1 1 3 ξ (x) = + (x − β) , x ∈ (a, b). 2 α

Corollary 2.9. Let X : Ω → (a, b) be a continuous random variable and let q1 (x) be as in Proposition 2.5. The random variable X has pd f (1.1.40) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) 3 (x − β)2 = 1 , 3 ξ (x) q1 (x) − q2 (x) α − (x − β)

x ∈ (a, b).

Corollary 2.10. The general solution of the differential equation in Corollary 2.9 is 1 ξ (x) = − (x − β)3 α 

where D is a constant.

−1  Z  2 −1 − 3 (x − β) (q1 (x)) q2 (x) dx + D ,

Proposition 2.6. Let X  : Ω → (0, ∞) be a continuous random variable and let q1 (x) =    −β p    −δ −β x −δ  exp − log 1− 1+( λ ) 1− 1+( λx )
−β x −δ and q (x) = q (x) 1 + for x > 0.      2 1 α−1 p  λ −δ −β − log 1− 1+( λx ) Then, the random variable X has pd f (1.1.42) if and only if the function ξ defined in Theorem 2.1 is of the form ( )   x −δ −β 1 ξ (x) = 1+ 1+ , x > 0. 2 λ

Characterizations of Distributions

293

Corollary 2.11. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.6. The random variable X has pd f (1.1.42) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation 
−β−1 x −δ δ −δ−1 βδλ x 1 + λ 1 (x) =− , −β 
ξ (x) q1 (x) − q2 (x) x −δ 1− 1+ λ ξ0 (x) q

x > 0.

Corollary 2.12. The general solution of the differential equation in Corollary 2.11 is "



ξ (x) = 1 − 1 +

 x −δ −β

#−1

× λ # "Z   x −δ −β−1 −1 δ −δ−1 (q1 (x)) q2 (x) dx + D , βδλ x 1+ λ

where D is a constant. Proposition 2.7. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = (1 + βH (x; η))−1 and q2 (x) = q1 (x) e−θH(x;η) for x > 0. Then, the random variable X has pd f (1.1.46) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−θH(x;η) , x > 0. 2 Corollary 2.13. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.7. The random variable X has pd f (1.1.46) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = θh (x; η) , ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.14. The general solution of the differential equation in Corollary 2.13 is  Z  −1 θH(x;η) −θH(x;η) ξ (x) = e − θh (x; η) e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.8. Let X : Ω → (1, ∞) be a continuous random variable and let q1 (x) =
a b x axb − b + 1 and q2 (x) = q1 (x) e b (1−x ) for x > 1. Then, the random variable X has pd f (1.1.52) if and only if the function ξ defined in Theorem 2.1 is of the form b 1 a ξ (x) = e b (1−x ) , x > 1. 2

Corollary 2.15. Let X : Ω → (1, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.8. The random variable X has pd f (1.1.52) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation

G. G. Hamedani

294

ξ0 (x) q1 (x) = axb−1 , ξ (x) q1 (x) − q2 (x)

x > 1.

Corollary 2.16. The general solution of the differential equation in Corollary 2.15 is  Z  −1 − ab (1−xb ) b−1 ab (1−xb ) ξ (x) = e − ax e (q1 (x)) q2 (x) dx + D , where D is a constant.

Proposition 2.9. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  1−a C(θe−z ) 1− C(θ)  b−1 C(θe−z ) C0 (θe−z ) 1 C(θ)

and q2 (x) = q1 (x) e−z for x > 0. Then, the random variable X has

pd f (1.1.56) if and only if the function ξ defined in Theorem 2.1 is of the form β 2 1 ξ (x) = e−αx− 2 x , x > 0. 2

Corollary 2.17. Let X : Ω → (1, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.9. The random variable X has pd f (1.1.56) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = α + βx, ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.18. The general solution of the differential equation in Corollary 2.17 is  Z  −1 αx+ β2 x2 −αx− β2 x2 ξ (x) = e − (α + βx) e (q1 (x)) q2 (x) dx + D , where D is a constant.

Proposition 2.10. Let X : Ω → R be a continuous random variable and let q1 (x) = −α 1 G(x;β)α and q2 (x) = q1 (x) e− α G(x;β) for x ∈ R. Then, the random variable X has 1−αG(x;β)α pd f (1.1.80) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

o −α 1 1n 1 + e− α G(x;β) , x ∈ R. 2

Corollary 2.19. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.10. The random variable X has pd f (1.1.80) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation 1

−α

ξ0 (x) q1 (x) g (x; β) G (x; β)−α−1 e− α G(x;β) = , −α 1 ξ (x) q1 (x) − q2 (x) 1 − e− α G(x;β)

x ∈ R.

Characterizations of Distributions

295

Corollary 2.20. The general solution of the differential equation in Corollary 2.19 is o n −α −1 1 × ξ (x) = 1 − e− α G(x;β)   Z −α−1 − α1 G(x;β)−α −1 − g (x; β)G (x; β) e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.11. Let X : Ω → R be a continuous random variable and let q1 (x) =  h   i  −1  G(x;η) β h iβ  C0 θ 1−exp −α G(x;η)  and q2 (x) = q1 (x) exp −α G(x;η)  for x ∈ R. Then, C(θ) G(x;η)

the random variable X has pd f (1.1.84) if and only if the function ξ defined in Theorem 2.1 is of the form (   ) G (x; η) β 1 ξ (x) = exp −α , x ∈ R. 2 G (x; η)

Corollary 2.21. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.11. The random variable X has pd f (1.1.84) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) (G (x; η))β−1 = αβg (x; η)
β+1 , ξ (x) q1 (x) − q2 (x) G (x; η)

x ∈ R.

Corollary 2.22. The general solution of the differential equation in Corollary 2.21 is   β−1 R (  β ) − αβg (x; η) (G(x;η)) β+1 × G (x; η) (G(x;η))    ξ (x) = exp α h iβ   , G(x;η) −1 G (x; η) exp −α G(x;η) (q1 (x)) q2 (x) dx + D

where D is a constant.

Proposition 2.12. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = 

h i 3 h i β α β 1−α 1− 1−e−x 1−e−x e

      β α β α − 1−e−x / 1− 1−e−x

β

and q2 (x) = q1 (x) e−x for x > 0. Then, the random variable X

has pd f (1.1.88) if and only if the function ξ defined in Theorem 2.1 is of the form 1 β ξ (x) = e−x , x > 0. 2 Corollary 2.23. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.12. The random variable X has pd f (1.1.88) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = βxβ−1 , ξ (x) q1 (x) − q2 (x)

x > 0.

G. G. Hamedani

296

Corollary 2.24. The general solution of the differential equation in Corollary 2.23 is  Z  −1 xβ β−1 −xβ ξ (x) = e − βx e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.13. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = h i β β −1 β 1 − 2λe−αx + 3λe−2αx and q2 (x) = q1 (x) e−αx for x > 0. Then, the random variable X has pd f (1.1.100) if and only if the function ξ defined in Theorem 2.1 is of the form 1 β ξ (x) = e−αx , x > 0. 2 Corollary 2.25. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.13. The random variable X has pd f (1.1.100) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = αβxβ−1 , ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.26. The general solution of the differential equation in Corollary 2.25 is  Z  −1 αxβ β−1 −αxβ ξ (x) = e − αβx e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.14. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) =  h i h i2 −1 b b a a λ1 + 2 (λ2 − λ1 ) 1 − (1 − x ) + 3 (1 − λ2 ) λ1 + 2 (λ2 − λ1 ) 1 − (1 − x ) and

q2 (x) = q1 (x) (1 − xa ) for x ∈ (0, 1). Then, the random variable X has pd f (1.1.102) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

b (1 − xa ), x ∈ (0, 1). b+1

Corollary 2.27. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.14. The random variable X has pd f (1.1.102) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) abxa−1 = , ξ (x) q1 (x) − q2 (x) 1 − xa

x ∈ (0, 1).

Corollary 2.28. The general solution of the differential equation in Corollary 2.27 is  Z  β ξ (x) = (1 − xa )−1 − abxa−1e−αx (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

Characterizations of Distributions

297

Proposition 2.15. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = i−1 h k k k and q2 (x) = q1 (x) e(x/λ) for x ∈ (1 − λ1 − λ2 ) e2(x/λ) + 2 (λ1 + 2λ2 ) e(x/λ) − 3λ2

(0, ∞). Then, the random variable X has pd f (1.1.104) if and only if the function ξ defined in Theorem 2.1 is of the form k 2 ξ (x) = e(x/λ) , x ∈ (0, ∞) . 3

Corollary 2.29. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.15. The random variable X has pd f (1.1.104) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = −2kλ−k xk−1 , ξ (x) q1 (x) − q2 (x)

x ∈ (0, ∞) .

Corollary 2.30. The general solution of the differential equation in Corollary 2.29 is Z  −1 −(x/λ)k −k k−1 (x/λ)k ξ (x) = e 2kλ x e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.16. Let X : Ω → R be a continuous random variable and let q1 (x) =  −1 1+τh(x) and q2 (x) = q1 (x) G (x) for xR. Then, the random variable X has pd f 1+κτ (1.1.110) if and only if the function ξ defined in Theorem 2.1 is of the form 1 {1 + G (x)}, x ∈ R. 2 Corollary 2.31. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.16. The random variable X has pd f (1.1.110) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ (x) =

g (x) ξ0 (x) q1 (x) = , ξ (x) q1 (x) − q2 (x) 1 − G (x)

x ∈ R.

Corollary 2.32. The general solution of the differential equation in Corollary 2.31 is Z  −1 −1 ξ (x) = {1 − G (x)} g (x) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.17. Let X : Ω → (0, β) be a continuous random variable and let q1 (x) = h  α i1−θ  αθ 1+γ−γ βx x  α for x ∈ (0, β). Then, the random variable X has and q (x) = q (x) 2 1 β x 1+γ−2γ

β

pd f (1.1.124) if and only if the function ξ defined in Theorem 2.1 is of the form (  αθ ) 1 x 1+ , x ∈ (0, β). ξ (x) = 2 β

G. G. Hamedani

298

Corollary 2.33. Let X : Ω → (0, β) be a continuous random variable and let q1 (x) be as in Proposition 2.17. The random variable X has pd f (1.1.124) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation  αθ−1 x αθ β 1 (x) =   αθ  , ξ (x) q1 (x) − q2 (x) β 1 − βx ξ0 (x) q

x ∈ (0, β).

Corollary 2.34. The general solution of the differential equation in Corollary 2.33 is #  αθ #−1 " Z  αθ−1 x αθ x −1 ξ (x) = 1 − − (q1 (x)) q2 (x) dx + D , β β β "

where D is a constant. Proposition 2.18. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = −1      α α ε(1−eβx)−γεx α(1−eβx)−γδx β and q2 (x) = q1 (x) e β for x > 0. δ(1 − λ) + λ (δ + ε) e

Then, the random variable X has pd f (1.1.130) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e 2

  α βx β α(1−e )−γδx

, x > 0.

Corollary 2.35. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.18. The random variable X has pd f (1.1.130) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation   ξ0 (x) q1 (x) = δ γ + αeβx , ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.36. The general solution of the differential equation in Corollary 2.35 is  

− α α(1−eβx)+γδx ξ (x) = e β ×  Z    α α(1−eβx)−γδx − δ γ + αeβx e β (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

Proposition 2.19. Let X : Ω → R be a continuous random variable and let q1 (x) =  n oβ a β+1 [1−G(x;γ) ] − log(1−G(x;γ)a ) and q2 (x) = q1 (x) G (x; γ)a for x ∈ R. Then, exp α a a β 1−G(x;γ) {− log(1−G(x;γ) )}

the random variable X has pd f (1.1.132) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 {1 + G (x; γ)a } , x ∈ R. 2

Characterizations of Distributions

299

Corollary 2.37. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.19. The random variable X has pd f (1.1.132) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) ag (x; γ)G (x; γ)a−1 = , ξ (x) q1 (x) − q2 (x) 1 − G (x; γ)a

x ∈ R.

Corollary 2.38. The general solution of the differential equation in Corollary 2.37 is a −1

ξ (x) = {1 − G (x; γ) } where D is a constant.

 Z  a−1 −1 − ag (x; γ) G (x; γ) (q1 (x)) q2 (x) dx + D ,

Remark 2.1. Similar characterizations can be stated for the distributions (1.1.133), (1.1.135) and (1.1.137). Proposition 2.20. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = −1 
b
b α − θ αb +1 xb+α e−θxα and q2 (x) = q1 (x) x−1 for x > 0. Then, + θ + 1 γ + 1, θx α α the random variable X has pd f (1.1.142) if and only if the function ξ defined in Theorem 2.1 is of the form b −1 x , x > 0. b+1 Corollary 2.39. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.20. The random variable X has pd f (1.1.142) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ (x) =

ξ0 (x) q1 (x) = bx−1 , ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.40. The general solution of the differential equation in Corollary 2.39 is  Z  −1 b −(b+1) ξ (x) = x − bx (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.21. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =
−1 λx−1 1 + x−1 e and q2 (x) = q1 (x) x−1 for x > 0. Then, the random variable X has pd f (1.1.146) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

φ −1 x , x > 0. φ+1

Corollary 2.41. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.21. The random variable X has pd f (1.1.146) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = φx−1 , ξ (x) q1 (x) − q2 (x)

x > 0.

G. G. Hamedani

300

Corollary 2.42. The general solution of the differential equation in Corollary 2.41 is  Z  −1 φ −(φ+1) ξ (x) = x − φx (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.22. Let X i: Ω → (0, 1) be a continuous random variable and let q1 (x) = h λ exp − 2µ2 log x (logx + µ)2 (− log x)5/2 and q2 (x) = q1 (x) (− log x) for 0 < x < 1. Then, the random variable X has pd f (1.1.148) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

2 (− logx) , x0 < x < 1. 3

Corollary 2.43. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.22. The random variable X has pd f (1.1.148) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = 2x−1 (− log x)−1 , ξ (x) q1 (x) − q2 (x)

0 < x < 1.

Corollary 2.44. The general solution of the differential equation in Corollary 2.43 is  Z  −1 −1 −1 ξ (x) = (− logx) − 2x (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.23. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = h θx/σ i−1 θ(σ+1) e αx2 eθx + and q2 (x) = q1 (x) e− σ x for x > 0. Then, the random variable X 1+αx2 σ(σ2 +αx2 ) has pd f (1.1.160) if and only if the function ξ defined in Theorem 2.1 is of the form 1 θ(σ+1) ξ (x) = e− σ x , x > 0. 2 Corollary 2.45. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.23. The random variable X has pd f (1.1.160) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) θ (σ + 1) = , ξ (x) q1 (x) − q2 (x) σ

x > 0.

Corollary 2.46. The general solution of the differential equation in Corollary 2.45 is  Z  θ(σ+1) θ (σ + 1) − θ(σ+1) −1 x σ σ ξ (x) = e − e (q1 (x)) q2 (x) dx + D , σ where D is a constant.

Characterizations of Distributions

301

Proposition 2.24. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = {1 − β [1 − Bc,k,η (x)]}s+1 and q2 (x) = q1 (x) Bc,k,η (x) for x > 0. Then, the random variable X has pd f (1.1.162) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 {1 + Bc,k,η (x)} , x > 0. 2

Corollary 2.47. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.24. The random variable X has pd f (1.1.162) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation bc,k,η (x) ξ0 (x) q1 (x) = , ξ (x) q1 (x) − q2 (x) 1 − Bc,k,η (x)

x > 0.

Corollary 2.48. The general solution of the differential equation in Corollary 2.47 is  Z  ξ (x) = {1 − Bc,k,η (x)}−1 − bc,k,η (x) (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

Proposition 2.25. Let X :Ω → (0, ∞) be a continuous random variable and let q1 (x) =  h ib h ib exp a 1 − (1 + xα )−2β and q2 (x) = q1 (x) 1 − (1 + xα )−2β for x > 0. Then, the

random variable X has pd f (1.1.178) if and only if the function ξ defined in Theorem 2.1 is of the form  h ib  1 α −2β 1 + 1 − (1 + x ) , x > 0. ξ (x) = 2

Corollary 2.49. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.25. The random variable X has pd f (1.1.178) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation h ib−1 α−1 (1 + xα )−2β−1 1 − (1 + xα )−2β 2bαβx ξ (x) q1 (x) = , h ib ξ (x) q1 (x) − q2 (x) 1 − 1 − (1 + xα )−2β 0

x > 0.

Corollary 2.50. The general solution of the differential equation in Corollary 2.49 is 

h

α −2β

ib −1

ξ (x) = 1 − 1 − (1 + x ) ×  Z  h ib−1 −2β−1 −2β − 2bαβxα−1 (1 + xα ) 1 − (1 + xα ) (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

G. G. Hamedani

302

Proposition 2.26. Let X : Ω → R be a continuous random variable and let q1 (x) = h i−1
j j i and q2 (x) = q1 (x) G (x) for x ∈ R. Then, the ∑ki=0 ∑ij=0 (−1) (1 + j) j λi [G (x)] random variable X has pd f (1.1.180) if and only if the function ξ defined in Theorem 2.1 is of the form

1 {1 + G (x)}, x ∈ R. 2 Corollary 2.51. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.26. The random variable X has pd f (1.1.180) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ (x) =

ξ0 (x) q1 (x) g (x) = , ξ (x) q1 (x) − q2 (x) 1 − G (x)

x ∈ R.

Corollary 2.52. The general solution of the differential equation in Corollary 2.51 is  Z  −1 −1 ξ (x) = {1 − G (x)} − g (x) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.27. Let X : Ω → (0, ∞) be a continuous random variable and  n o−1  1−θ   β −β β −β β −β β −β θ let q1 (x) = 1 − e−θbδ x 2 − e−bδ x exp ae−θbδ x 2 − e−bδ x and β −β

q2 (x) = q1 (x) e−θbδ x for x > 0. Then, the random variable X has pd f (1.1.182) if and only if the function ξ defined in Theorem 2.1 is of the form

o β −β 1n 1 + e−θbδ x , x > 0. 2 Corollary 2.53. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.27. The random variable X has pd f (1.1.182) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ (x) =

β −β

ξ0 (x) q1 (x) θbβδβ x−(β+1) e−θbδ x = , x > 0. ξ (x) q1 (x) − q2 (x) 1 − e−θbδβ x−β Corollary 2.54. The general solution of the differential equation in Corollary 2.53 is n

−θbδβ x−β

ξ (x) = 1 − e

where D is a constant.

 o−1  Z −1 β −(β+1) −θbδβ x−β (q1 (x)) q2 (x) dx + D , − θbβδ x e

Proposition 2.28. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =
x 1−a and q2 (x) = q1 (x) (1 + x)−1 for x > 0. Then, the random variable X has pd f 1+x (1.1.194) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

b (1 + x)−1 , x > 0. b+1

Characterizations of Distributions

303

Corollary 2.55. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.28. The random variable X has pd f (1.1.194) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = b (1 + x)−1 , ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.56. The general solution of the differential equation in Corollary 2.55 is  Z  b −b−1 −1 ξ (x) = (1 + x) − b (1 + x) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.29. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = n o−1 θ3 (1 + x)ab+1 e−θx + (1 + θ) xa−1 (1 + x)−1 and q2 (x) = q1 (x) (1 + x)−1 for x > 0. Then, the random variable X has pd f (1.1.196) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

a+b (1 + x)−1 , x > 0. a+b+1

Corollary 2.57. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.29. The random variable X has pd f (1.1.196) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = (a + b) (1 + x)−1 , ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.58. The general solution of the differential equation in Corollary 2.57 is  Z  a+b −a−b−1 −1 ξ (x) = (1 + x) − (a + b) (1 + x) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.30. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = 
α
2α −1
1 − 2λ xx0 + 3λ xx0 and q2 (x) = q1 (x) xx0 for x > 0. Then, the random variable X has pd f (1.1.198) if and only if the function ξ defined in Theorem 2.1 is of the form α  x0  ξ (x) = , x > 0. α+1 x Corollary 2.59. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.30. The random variable X has pd f (1.1.198) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = αx−1 , ξ (x) q1 (x) − q2 (x)

x > 0.

G. G. Hamedani

304

Corollary 2.60. The general solution of the differential equation in Corollary 2.59 is  Z  −1 α −α−1 ξ (x) = x − αx (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.31. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = τ π x−1 [ν2 sinh2 (log x)+1] + arctan [ν sinh(logx)] and q2 (x) = q1 (x) x−1 for x > 0. Then, exp 2 2 cosh(log x) the random variable X has pd f (1.1.206) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = x−1 , x > 0. 2 Corollary 2.61. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.31. The random variable X has pd f (1.1.206) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = x−1 , ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.62. The general solution of the differential equation in Corollary 2.61 is  Z  −1 −2 ξ (x) = x − x (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.32. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  −−b n  oa+b  a k k k e−kβx −λx 1 − (1 − c) 1 − e−kβx −λx and q2 (x) = q1 (x) 1 − e−kβx −λx for

x > 0. Then, the random variable X has pd f (1.1.208) if and only if the function ξ defined in Theorem 2.1 is of the form  a o 1n k 1 + 1 − e−kβx −λx , x > 0. 2 Corollary 2.63. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.32. The random variable X has pd f (1.1.208) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ (x) =

a−1
−kβxk −λx  k−1 −kβxk −λx a βkx + λ e 1 − e 1 (x) =
a , ξ (x) q1 (x) − q2 (x) 1 − 1 − e−kβxk −λx ξ0 (x) q

x > 0.

Corollary 2.64. The general solution of the differential equation in Corollary 2.63 is n  a o−1 k ξ (x) = 1 − 1 − e−kβx −λx ×  Z     a−1 −1 k−1 −kβxk −λx −kβxk −λx − a βkx + λ e 1−e (q1 (x)) q2 (x) dx + D ,

Characterizations of Distributions

305

where D is a constant. Proposition 2.33. Let X : Ω → R be a continuous random variable and let q1 (x) = 2   [1−log[G(x;η)]] and q (x) = q (x) G (x; η) for x ∈ R. Then, the random variable 2 1 {θ[1−log[G(x;η)]]+1} X has pd f (1.1.218) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

θ G (x; η) , x ∈ R. θ+1

Corollary 2.65. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.33. The random variable X has pd f (1.1.218) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation θg (x; η) ξ0 (x) q1 (x) = , ξ (x) q1 (x) − q2 (x) G (x; η)

x ∈ R.

Corollary 2.66. The general solution of the differential equation in Corollary 2.65 is  Z   −1 −1 ξ (x) = G (x; η) − θg (x; η) (q1 (x)) q2 (x) dx + D , where D is a constant.

Proposition 2.34. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) =
−1 θ − xβ and q2 (x) = q1 (x) xσ for 0 < x < 1. Then, the random variable X has pd f (1.1.222) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 (1 + xσ ), 0 < x < 1. 2

Corollary 2.67. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.34. The random variable X has pd f (1.1.222) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) σxσ−1 = , ξ (x) q1 (x) − q2 (x) 1 − xσ

0 < x < 1.

Corollary 2.68. The general solution of the differential equation in Corollary 2.67 is  Z  −1 σ −1 σ−1 ξ (x) = {1 − x } − σx (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.35. Let X : Ω → (0, ∞) be a
continuous random variable and let q1 (x) ≡ 1 a k+1 and q2 (x) = exp − k+1 x + bc (ecx − 1) for x > 0. Then, the random variable X has pd f (1.1.234) if and only if the function ξ defined in Theorem 2.1 is of the form    1 a k+1 b cx ξ (x) = exp − x + (e − 1) , x > 0. 2 k+1 c

G. G. Hamedani

306

Corollary 2.69. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.35. The random variable X has pd f (1.1.234) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = axk + becx , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.70. The general solution of the differential equation in Corollary 2.69 is  a k+1 b cx ξ (x) = exp x + (e − 1) × k+1 c

  R a k+1 − axk + becx exp − k+1 x + bc (ecx − 1) × , (q1 (x))−1 q2 (x) dx + D 

where D is a constant. Proposition 2.36. Let X : Ω → R be a continuous random variable and let q1 (x) = 
−λ 1−α
−1 1 − 1 + ex/θ and q2 (x) = q1 (x) 1 + ex/θ for x ∈ R. Then, the random variable X has pd f (1.1.248) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

−1 λ  1 + ex/θ , x ∈ R. λ+1

Corollary 2.71. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.36. The random variable X has pd f (1.1.248) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation  −1 ξ0 (x) q1 (x) λ = ex/θ 1 + ex/θ , x ∈ R. ξ (x) q1 (x) − q2 (x) θ

Corollary 2.72. The general solution of the differential equation in Corollary 2.71 is   λ  Z λ  −λ−1 −1 x/θ x/θ x/θ ξ (x) = 1 + e − e 1+e (q1 (x)) q2 (x) dx + D , θ

where D is a constant.

Proposition 2.37. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = √  −1 β√xeλx α + β (1 + 2λx) eλx e and q2 (x) = q1 (x) e−α x for x > 0. Then, the random variable X has pd f (1.1.250) if and only if the function ξ defined in Theorem 2.1 is of the form √ 1 ξ (x) = e−α x , x > 0. 2

Characterizations of Distributions

307

Corollary 2.73. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.37. The random variable X has pd f (1.1.250) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = αx−1/2 , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.74. The general solution of the differential equation in Corollary 2.73 is  Z  √ √ −1 α x −1/2 −α x ξ (x) = e − αx e (q1 (x)) q2 (x) dx + D , where D is a constant.

Proposition 2.38. Let X : Ω → R be a continuous random variable and let  h  i−1  β   G(x;η) −λ G(x;η) G(x;η) G(x;η)β G(x;η) G(x;η) q1 (x) = λ + β G(x;η) exp α e + λ and β−2 G(x;η) G(x;η) G(x;η)

q2 (x) = q1 (x) G (x; η) for x ∈ R. Then, the random variable X has pd f (1.1.252) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 (1 + G (x; η)) , x ∈ R. 2

Corollary 2.75. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.38. The random variable X has pd f (1.1.252) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) g (x; η) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; η) Corollary 2.76. The general solution of the differential equation in Corollary 2.75 is  Z  −1 −1 ξ (x) = (1 − G (x; η)) − g (x; η) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.39. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = h   i−1 2 2 sinh α 1 − e−σx and q2 (x) = q1 (x) e−σx for x > 0. Then, the random variable

X has pd f (1.1.266) if and only if the function ξ defined in Theorem 2.1 is of the form 1 2 ξ (x) = e−σx , x > 0. 2

Corollary 2.77. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.39. The random variable X has pd f (1.1.266) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = 2σx, x > 0. ξ (x) q1 (x) − q2 (x)

G. G. Hamedani

308

Corollary 2.78. The general solution of the differential equation in Corollary 2.77 is  Z  −1 σx2 −σx2 − 2σxe (q1 (x)) q2 (x) dx + D , ξ (x) = e

where D is a constant.

Proposition 2.40. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = x1−α (1 + α + β + x) −1 and q2 (x) = q1 (x) (β + x)−1 for x > 0. Then, the random variable X has pd f (1.1.280) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 (β + x)−1 , x > 0. 2

Corollary 2.79. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.40. The random variable X has pd f (1.1.280) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = (β + x)−1 , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.80. The general solution of the differential equation in Corollary 2.79 is  Z  −2 −1 ξ (x) = (β + x) − (β + x) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.41. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =    fy QY (1+x−c)

−k

ft (QY ((1+x−c )−k ))

−k

and q2 (x) = q1 (x) (1 + x−c )

for x > 0. Then, the random variable X

has pd f (1.1.288) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =


−k o 1n 1 + 1 + x−c , x > 0. 2

Corollary 2.81. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.41. The random variable X has pd f (1.1.288) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation −k−1

ξ0 (x) q1 (x) ckx−c−1 (1 + x−c ) = ξ (x) q1 (x) − q2 (x) 1 − (1 + x−c )−k

, x > 0.

Corollary 2.82. The general solution of the differential equation in Corollary 2.81 is  Z  n
o−1
−1 −c −k −c−1 −c −k−1 ξ (x) = 1 − 1 + x − ckx 1+x (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Characterizations of Distributions

309

Proposition 2.42. Let X : Ω → R be a continuous random variable and let q1 (x) = 2 {Φτ (x)+(1−ν)[1−Φ(x)]τ } −k and q2 (x) = q1 (x) (1 + x−c ) for x > 0. Then, the random variτ−1 Φ (x) able X has pd f (1.1.296) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

τ {1 − Φ (x)}, x ∈ R. τ+1

Corollary 2.83. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.42. The random variable X has pd f (1.1.296) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation τφ (x) ξ0 (x) q1 (x) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − Φ (x)

Corollary 2.84. The general solution of the differential equation in Corollary 2.83 is  Z  −1 −1 ξ (x) = {1 − Φ (x)} − τφ (x) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.43. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = 2   {1−(1−a)exp[−bκαβ(x)]} and q2 (x) = q1 (x) exp −καβ (x) for x > 0. Then, the random exp[−(b−1)καβ (x)] exp[αx+β/x] variable X has pd f (1.1.298) if and only if the function ξ defined in Theorem 2.1 is of the form   1 exp −καβ (x) , x > 0. 2 Corollary 2.85. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.43. The random variable X has pd f (1.1.298) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation
  α + β/x2 exp −καβ (x) ξ0 (x) q1 (x)   , x > 0. =− ξ (x) q1 (x) − q2 (x) 1 − exp −καβ (x) ξ (x) =

Corollary 2.86. The general solution of the differential equation in Corollary 2.85 is    −1 ξ (x) = 1 − exp −καβ (x) × Z 
  −1 2 α + β/x exp −καβ (x) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.44. Let X : Ω → R be a continuous random variable and let q1 (x) = h  i−1 G(x) (1−G(x))2 k 1−G(x)    G(x) cosh aK 1−G(x)

and q2 (x) = q1 (x) G (x) for x ∈ R. Then, the random variable X has

pd f (1.1.312) if and only if the function ξ defined in Theorem 2.1 is of the form

G. G. Hamedani

310

ξ (x) =

1 {1 + G (x)}, x ∈ R. 2

Corollary 2.87. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.44. The random variable X has pd f (1.1.312) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) g (x) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x) Corollary 2.88. The general solution of the differential equation in Corollary 2.87 is  Z  −1 −1 ξ (x) = {1 − G (x)} − g (x) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.45. Let
X : Ω → R be a continuous random variable and let q1 (x) ≡ 1 and q2 (x) = π2 arctan eλx for x ∈ R. Then, the random variable X has pd f (1.1.314) if and only if the function ξ defined in Theorem 2.1 is of the form    1 2 ξ (x) = 1 + arctan eλx , x ∈ R. 2 π

Corollary 2.89. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.45. The random variable X has pd f (1.1.314) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation 2βeβx

π(1+e2βx) ξ0 (x) q1 (x)
, x ∈ R. = 2 ξ (x) q1 (x) − q2 (x) 1 − π arctan eλx

Corollary 2.90. The general solution of the differential equation in Corollary 2.89 is " Z #   −1 βx 2 2βe
(q1 (x))−1 q2 (x) dx + D , ξ (x) = 1 − arctan eλx − π π 1 + e2βx

where D is a constant.

Proposition 2.46. Let X : Ω → R be a continuous random variable and let q1 (x) =  −1 (logα)αG(x;η) − αeαG(x;η) and q2 (x) = q1 (x) G (x; η) for x ∈ R. Then, the random variable X has pd f (1.1.316) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 {1 + G (x; η)} , x ∈ R. 2

Characterizations of Distributions

311

Corollary 2.91. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.46. The random variable X has pd f (1.1.316) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation g (x; η) ξ0 (x) q1 (x) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; η)

Corollary 2.92. The general solution of the differential equation in Corollary 2.91 is  Z  ξ (x) = {1 − G (x; η)}−1 − g (x; η) (q1 (x))−1 q2 (x) dx + D ,

where D is a constant. Without loss of generality, we assume µ = 0 and σ = 1 in the cd f (1.1.327).

Proposition 2.47. Let X : Ω → R be a continuous random variable and let q1 (x) =  −1 φ [ax;µ,σ (α, β)] Φ [λax;µ,σ (α, β)] (x + β) and q2 (x) = q1 (x) x−1/2 for x ∈ R. Then, the random variable X has pd f (1.1.328) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = x−1/2 , x > 0. 2 Corollary 2.93. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.47. The random variable X has pd f (1.1.328) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) 1 = x−1 , x > 0. ξ (x) q1 (x) − q2 (x) 2

Corollary 2.94. The general solution of the differential equation in Corollary 2.93 is  Z  1 −3/2 ξ (x) = x1/2 − x (q1 (x))−1 q2 (x) dx + D , 2 where D is a constant.

Proposition h n  2.48. Let X :γΩ o → R bei a continuous random  variable and  let q1 (x) = β β α α e − 1 − 1 − G (x; η) (e − 1) and q2 (x) = q1 (x) 1 − G (x; η) for x ∈ R. The random variable X has pd f (1.1.330) if and only if the function ξ defined in Theorem 2.1 is of the form  γ  ξ (x) = 1 − G (x; η)β , x ∈ R. γ+1

Corollary 2.95. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.48. The random variable X has pd f (1.1.330) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) βγg (x; η) G (x; η)β−1 = , ξ (x) q1 (x) − q2 (x) 1 − G (x; η)β

x ∈ R.

G. G. Hamedani

312

Corollary 2.96. The general solution of the differential equation in Corollary 2.95 is  n o−1  Z β β−1 −1 ξ (x) = 1 − G (x; η) − βγg (x; η) G (x; η) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.49. Let X : Ω → R be a continuous random variable and let  1−α  α α  q1 (x) = G (x; φ) − log (1 − β) G (x; φ) + (1 − β) G (x; φ) G (x; φ)α + G (x; φ)α and q2 (x) = q1 (x) G (x; φ)α for x ∈ R. The random variable X has pd f (1.1.344) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =


1 1 + G (x; φ)α , x ∈ R. 2

Corollary 2.97. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.49. The random variable X has pd f (1.1.344) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) αg (x; φ)G (x; φ)α−1 = , ξ (x) q1 (x) − q2 (x) 1 − G (x; φ)α

x ∈ R.

Corollary 2.98. The general solution of the differential equation in Corollary 2.97 is α −1

ξ (x) = 1 − G (x; φ) 

where D is a constant.

 Z  α−1 −1 − αg (x; φ)G (x; φ) (q1 (x)) q2 (x) dx + D ,

Proposition 2.50. Let X : Ω → [1, ∞) be a continuous random variable and let q1 (x) = −β αx −1 and q2 (x) = q1 (x) x−1 for x > 1. The random variable X has pd f (1.1.366) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

β −1 x , x ∈ R. β+1

Corollary 2.99. Let X : Ω → [1, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.50. The random variable X has pd f (1.1.366) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = βx−1 , ξ (x) q1 (x) − q2 (x)

x > 1.

Corollary 2.100. The general solution of the differential equation in Corollary 2.99 is  Z  −1 β −β−1 ξ (x) = x − βx (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Characterizations of Distributions

313

Proposition 2.51. Let X : Ω → (0, ∞) be a continuous random variable and    
 1−δ 1+θ+θx −θx 1 e and q2 (x) = let q1 (x) = − log 1 − 1−e λ 1 − exp λ 1 − 1+θ 
 1+θ+θx −θx q1 (x) exp λ 1 − 1+θ e for x > 0. The random variable X has pd f (1.1.374) if and only if the function ξ defined in Theorem 2.1 is of the form     1 λ 1 + θ + θx −θx e + exp λ 1 − e , x > 0. ξ (x) = 2 1+θ Corollary 2.101. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.51. The random variable X has pd f (1.1.374) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = ξ (x) q1 (x) − q2 (x)


 λθ2 (1+x)e−θx −θx exp λ 1 − 1+θ+θx 1+θ e (1+θ) 
 , −θx eλ − exp λ 1 − 1+θ+θx 1+θ e

x > 0.

Corollary 2.102. The general solution of the differential equation in Corollary 2.101 is −1    1 + θ + θx −θx λ e × ξ (x) = e − exp λ 1 − 1+θ      Z λθ2 (1 + x) e−θx 1 + θ + θx −θx −1 − exp λ 1 − e (q1 (x)) q2 (x) dx + D , (1 + θ) 1+θ where D is a constant. Proposition 2.52. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = 1 x x(x+1) and q2 (x) = q1 (x) w for x > 0. The random variable X has pd f (1.1.376) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = wx , x > 0. 2 Corollary 2.103. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.52. The random variable X has pd f (1.1.376) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = − log(w) , ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.104. The general solution of the differential equation in Corollary 2.103 is  Z  −1 −x x ξ (x) = w − log(w) w (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.53. Let X : Ω → R be a continuous random variable and let h n oi1−a h n oi1−b λ+G(x) G(x) λ+G(x) G(x) q1 (x) = 1 − (1+λ)G(x) exp −λ G(x) exp −λ and q2 (x) = (1+λ)G(x) G(x)

G. G. Hamedani n o G(x) q1 (x) exp −λ G(x) for x ∈ R. The random variable X has pd f (1.1.378) if and only if the function ξ defined in Theorem 2.1 is of the form   1 G (x) ξ (x) = exp −λ , x ∈ R. 2 G (x) 314

Corollary 2.105. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.53. The random variable X has pd f (1.1.378) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) λg (x) = , ξ (x) q1 (x) − q2 (x) G (x)2

x ∈ R.

Corollary 2.106. The general solution of the differential equation in Corollary 2.105 is #   " Z  λg (x) G (x) G (x) exp −λ ξ (x) = exp λ − (q1 (x))−1 q2 (x) dx + D , G (x) G (x) G (x)2 where D is a constant. Proposition
2.54. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = λ2 + x − 2λ and q2 (x) = q1 (x) e−x/λ for x > 0. The random variable X has pd f (1.1.380) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−x/λ , x > 0. 2 Corollary 2.107. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.54. The random variable X has pd f (1.1.380) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) 1 = , ξ (x) q1 (x) − q2 (x) λ

x > 0.

Corollary 2.108. The general solution of the differential equation in Corollary 2.107 is   Z 1 −x/λ −1 x/λ ξ (x) = e − e (q1 (x)) q2 (x) dx + D , λ where D is a constant. Proposition 2.55. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = n oα+1 θx 1+ β1 (1−(1+ θ+1 )e−θx ) and q2 (x) = q1 (x) e−θx for x > 0. The random variable X has x+1 pd f (1.1.430) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−θx , x > 0. 2

Characterizations of Distributions

315

Corollary 2.109. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.55. The random variable X has pd f (1.1.430) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = θ, ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.110. The general solution of the differential equation in Corollary 2.109 is  Z  −1 θx −θx ξ (x) = e − θe (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.56. Let X : Ω → R be a continuous random variable and let q1 (x) = i2 1+G(x)λ  a−1   a b−1 1−G(x)λ 1−G(x)λ 1− λ λ h

1+G(x)

and q2 (x) = q1 (x) G (x) for x ∈ R. The random variable X

1+G(x)

has pd f (1.1.440) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

λ G(x) , x ∈ R. λ+1

Corollary 2.111. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.56. The random variable X has pd f (1.1.440) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) λg (x) , = ξ (x) q1 (x) − q2 (x) G (x)

x ∈ R.

Corollary 2.112. The general solution of the differential equation in Corollary 2.111 is  Z  −1 −1 ξ (x) = G(x) − λg (x) (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition h
2.57. Let i X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = xβ+1 α β −λx e and q2 (x) = q1 (x) e−λx for x > 0. The random variable X has pd f x β+λx exp (1.1.464) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−λx , x > 0. 2 Corollary 2.113. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.57. The random variable X has pd f (1.1.464) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = λ, ξ (x) q1 (x) − q2 (x)

x > 0.

316

G. G. Hamedani

Corollary 2.114. The general solution of the differential equation in Corollary 2.113 is  Z  ξ (x) = eλx − λe−λx (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.58. Let X : Ω → R be a continuous random variable and let q1 (x) = n o−1 1 − λ + 6λG (x) − 6λG (x)2 and q2 (x) = q1 (x) G (x) for x ∈ R. The random variable X has pd f (1.1.466) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 {1 + G (x)} , x ∈ R. 2

Corollary 2.115. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.58. The random variable X has pd f (1.1.466) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) g (x) = , ξ (x) q1 (x) − q2 (x) 1 − G (x)

x ∈ R.

Corollary 2.116. The general solution of the differential equation in Corollary 2.115 is  Z  −1 −1 ξ (x) = {1 − G (x)} − g (x) (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.59. Let X : Ω → R be a continuous random variable and let q1 (x) = 2 [α+β cos ( π G(x))+γ sin( π2 G(x))+ θ2 sin(πG(x)) ]  2  and q2 (x) = q1 (x) G (x) for x ∈ R. The random vari3 β+α cos( π2 G(x))+θ sin( π2 G(x)) able X has pd f (1.1.470) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 {1 + G (x)} , x ∈ R. 2

Corollary 2.117. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.59. The random variable X has pd f (1.1.470) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) g (x) = , ξ (x) q1 (x) − q2 (x) 1 − G (x)

x ∈ R.

Corollary 2.118. The general solution of the differential equation in Corollary 2.117 is  Z  ξ (x) = {1 − G (x)}−1 − g (x) (q1 (x))−1 q2 (x) dx + D , where D is a constant.

Characterizations of Distributions

317

Proposition 2.60. Let X : Ω → R be a continuous random variable and let q1 (x) = 
2 (α − 1) θ − (1 − θ) αG(x) − 1 and q2 (x) = q1 (x) αG(x) for x ∈ R. The random variable X has pd f (1.1.478) for α 6= 1, if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

o 1n α + αG(x) , x ∈ R. 2

Corollary 2.119. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.60. The random variable X has pd f (1.1.478) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) log(α) g (α) (x) αG(x) = , ξ (x) q1 (x) − q2 (x) α − αG(x)

x ∈ R.

Corollary 2.120. The general solution of the differential equation in Corollary 2.119 is  n o−1  Z −1 G(x) G(x) ξ (x) = α − α − log(α) (x) α (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition
2.61. Let X : Ω → R be a continuous random variable and let q1 (x) = 1 + e2βx e−2βx and q2 (x) = q1 (x) e−βx for x ∈ R. The random variable X has pd f (1.1.482) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−βx , x ∈ R. 2 Corollary 2.121. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.61. The random variable X has pd f (1.1.482) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = β, ξ (x) q1 (x) − q2 (x)

x ∈ R.

Corollary 2.122. The general solution of the differential equation in Corollary 2.121 is  Z  −1 βx −βx ξ (x) = e − βe (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.62. Let X : Ω → R be a continuous random variable and let q1 (x) = (1+eπx)e−πx and q2 (x) = q1 (x) e−πx/2 for x ∈ R. The random variable X has pd f (1.1.484) [(1−αx)2 +1] if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−πx/2 , x ∈ R. 2

G. G. Hamedani

318

Corollary 2.123. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.62. The random variable X has pd f (1.1.484) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation π ξ0 (x) q1 (x) = , ξ (x) q1 (x) − q2 (x) 2

x ∈ R.

Corollary 2.124. The general solution of the differential equation in Corollary 2.123 is  Z  π −πx/2 −1 πx/2 ξ (x) = e − e (q1 (x)) q2 (x) dx + D , 2 where D is a constant. Proposition 2.63. Let X : Ω → R be a continuous random variable and let q1 (x) =
2 1 + e−λG(x) and q2 (x) = q1 (x) e−λG(x) for x ∈ R. The random variable X has pd f (1.1.486) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

 1  −λG(x) e + e−λ , x ∈ R. 2

Corollary 2.125. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.63. The random variable X has pd f (1.1.486) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) λg (x) = −λG(x) , ξ (x) q1 (x) − q2 (x) e − e−λ

x ∈ R.

Corollary 2.126. The general solution of the differential equation in Corollary 2.125 is   −1  Z −1 −λG(x) −λ ξ (x) = e −e − λg (x) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.64. LetoX : Ω → R be a continuous random variable and let q1 (x) =  n exp − (1 − Φ (x))−λ /λ (1 − Φ (x))λ+1 and q2 (x) = q1 (x) Φ (x) for x ∈ R. The random variable X has pd f (1.1.488) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 (1 + Φ (x)) , x ∈ R. 2

Corollary 2.127. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.64. The random variable X has pd f (1.1.488) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) φ (x) = , ξ (x) q1 (x) − q2 (x) 1 − Φ (x)

x ∈ R.

Characterizations of Distributions

319

Corollary 2.128. The general solution of the differential equation in Corollary 2.127 is  Z  ξ (x) = (1 − Φ (x))−1 − φ (x) (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.65. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = h  i1−a β β − log 1 − e−αx and q2 (x) = q1 (x) e−αx for x > 0. The random variable X has pd f (1.1.500) if and only if the function ξ defined in Theorem 2.1 is of the form 1 β ξ (x) = e−αx , x > 0. 2 Corollary 2.129. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.65. The random variable X has pd f (1.1.500) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = αβxβ−1 , ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.130. The general solution of the differential equation in Corollary 2.129 is   Z −1 β−1 −αxβ αxβ ξ (x) = e − αβx e (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.66. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = (1+ex )3 e−2x (1−λ)ex +3λ+1

and q2 (x) = q1 (x) e−x for x > 0. The random variable X has pd f (1.1.512) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−x , x > 0. 2

Corollary 2.131. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.66. The random variable X has pd f (1.1.512) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = 1, ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.132. The general solution of the differential equation in Corollary 2.131 is  Z  −1 x −x ξ (x) = e − e (q1 (x)) q2 (x) dx + D , where D is a constant.

G. G. Hamedani

320

Proposition 2.67. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  1−α  
−1 θx(θ2 x2 +3θx+6) −θx e and q2 (x) = q1 (x) e−θx for x > 0. The ranθ + x3 1− 1+ θ4 +6

dom variable X has pd f (1.1.520) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−θx , x > 0. 2 Corollary 2.133. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.67. The random variable X has pd f (1.1.520) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = θ, ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.134. The general solution of the differential equation in Corollary 2.133 is  Z  −1 θx −θx ξ (x) = e − θe (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.68. Let X : Ω → R be a continuous random variable and let q1 (x) =  3 1 − G (x; ϕ)α and q2 (x) = q1 (x) G (x; ϕ)α for x ∈ R. The random variable X has pd f (1.1.522) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 1 + G (x; ϕ)α , x ∈ R. 2

Corollary 2.135. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.68. The random variable X has pd f (1.1.522) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) αg (x; ϕ) G (x; ϕ)α−1 , = ξ (x) q1 (x) − q2 (x) 1 − G (x; ϕ)α

x ∈ R.

Corollary 2.136. The general solution of the differential equation in Corollary 2.135 is  Z   −1 ξ (x) = 1 − G (x; ϕ)α − αg (x; ϕ) G (x; ϕ)α−1 (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

Proposition 2.69. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =
−1 xr + θ2 xr+2 and q2 (x) = q1 (x) e−θx for x > 0. The random variable X has pd f (1.1.534) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−θx , x > 0. 2

Characterizations of Distributions

321

Corollary 2.137. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.69. The random variable X has pd f (1.1.534) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = θ, ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.138. The general solution of the differential equation in Corollary 2.137 is  Z  −1 θx −θx ξ (x) = e − θe (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.70. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =
−1 x + θ2 x3 and q2 (x) = q1 (x) e−θx for x > 0. The random variable X has pd f (1.1.536) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−θx , x > 0. 2 Corollary 2.139. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.70. The random variable X has pd f (1.1.536) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = θ, ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.140. The general solution of the differential equation in Corollary 2.139 is  Z  −1 θx −θx ξ (x) = e − θe (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.71. Let X : Ω → (0, 2π) be a continuous random variable and let q1 (x) =  −1 
e−2πλ λx2 −2π 1 + 2 + 2πλ (π − x) e + (x + π) and q2 (x) = q1 (x) e−λx for x > 0. (1−e−2πλ ) The random variable X has pd f (1.1.538) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−λx , x ∈ (0, 2π). 2 Corollary 2.141. Let X : Ω → (0, 2π) be a continuous random variable and let q1 (x) be as in Proposition 2.71. The random variable X has pd f (1.1.538) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = λ, ξ (x) q1 (x) − q2 (x)

x > 0.

G. G. Hamedani

322

Corollary 2.142. The general solution of the differential equation in Corollary 2.141 is  Z  ξ (x) = eλx − λe−λx (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.72. n Letγ oX : Ω → (θ, ∞) be a continuous random variable and let q1 (x) = [(γ−1)x+θ]−1 (x−θ) and q2 (x) = q1 (x) x−1 for x > θ. The random variable X has pd f exp γ−1 τx (x−θ)

(1.1.540) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = x−1 , x > θ. 2

Corollary 2.143. Let X : Ω → (θ, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.72. The random variable X has pd f (1.1.540) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = x−1 , ξ (x) q1 (x) − q2 (x)

x > θ.

Corollary 2.144. The general solution of the differential equation in Corollary 2.143 is  Z  ξ (x) = x − x−2 (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.73. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) = 2 [xb (1−m)b +mb (1−x)b ] and q2 (x) = q1 (x) xb for 0 < x < 1. The random variable X has pd f b−1 (1−x)

(1.1.542) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

 1 1 + xb , 0 < x < 1. 2

Corollary 2.145. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.73. The random variable X has pd f (1.1.542) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) bxb−1 = , ξ (x) q1 (x) − q2 (x) 1 − xb

0 < x < 1.

Corollary 2.146. The general solution of the differential equation in Corollary 2.145 is   −1  Z ξ (x) = 1 − xb − bxb−1 (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

Characterizations of Distributions

323

Proposition 2.74. Let X : Ω → R be a continuous random variable and let q1 (x) ≡ 1 and


q2 (x) = Φ ν1 γ + sinh π2 x , x ∈ R. The random variable X has pd f (1.1.544) if and only if the function ξ defined in Theorem 2.1 is of the form     π  1 1 ξ (x) = 1+Φ γ + sinh x , x ∈ R. 2 ν 2

Corollary 2.147. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.74. The random variable X has pd f (1.1.544) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation
1


π π π ξ0 (x) 2ν cosh 2 x φ ν γ + sinh 2 x


= , x ∈ R. ξ (x) − q2 (x) 1 − Φ 1ν γ + sinh π2 x

Corollary 2.148. The general solution of the differential equation in Corollary 2.147 is     π −1 1 γ + sinh x × ξ (x) = 1 − Φ ν 2  Z  π  1   π  π − cosh x φ γ + sinh x q2 (x) dx + D , 2ν 2 ν 2

where D is a constant.

Proposition 2.75. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =
−1 2x4 + θx2 + 2θ and q2 (x) = q1 (x) e−θx , x > 0. The random variable X has pd f (1.1.552) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−θx , x > 0. 2 Corollary 2.149. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.75. The random variable X has pd f (1.1.552) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = θ, ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.150. The general solution of the differential equation in Corollary 2.149 is  Z  ξ (x) = eθx − θe−θx (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.76. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) = λx (1 − x)2 and q2 (x) = q1 (x) e− 1−x , 0 < x < 1. The random variable X has pd f (1.1.570) if and only if the function ξ defined in Theorem 2.1 is of the form 1 λx ξ (x) = e− 1−x , 0 < x < 1. 2

G. G. Hamedani

324

Corollary 2.151. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.76. The random variable X has pd f (1.1.570) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = λ (1 − x)−2 , ξ (x) q1 (x) − q2 (x)

0 < x < 1.

Corollary 2.152. The general solution of the differential equation in Corollary 2.151 is  Z  λx λx −2 − 1−x −1 1−x ξ (x) = e − λ (1 − x) e (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.77. Let X : Ω → R be a continuous random variable and let q2 (x) = h i−1 b b (1 + λ) (1 + e−x ) − 2λ and q1 (x) = q2 (x) (1 + e−x ) , x ∈ R. The random variable X has pd f (1.1.574) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =


−b o 1n 1 + 1 + e−x , x ∈ R. 2

Corollary 2.153. Let X : Ω → R be a continuous random variable and let q2 (x) be as in Proposition 2.77. The random variable X has pd f (1.1.574) if and only if there exist functions q1 and ξ defined in Theorem 2.1 satisfying the following differential equation −b−1

be−x (1 + e−x ) ξ0 (x) q1 (x) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − (1 + e−x )−b Corollary 2.154. The general solution of the differential equation in Corollary 2.153 is  Z  n
o−1
−1 −x −b −x −x −b−1 ξ (x) = 1 − 1 + e − be 1+e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.78. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = 

  ν+1 β α p+(1−p) 1−e−(λx)   β α−1 1−e−(λx)

β

and q2 (x) = q1 (x) e−(λx) , x > 0. The random variable X has pd f

(1.1.576) if and only if the function ξ defined in Theorem 2.1 is of the form β 1 ξ (x) = e−(λx) , x > 0. 2

Corollary 2.155. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.78. The random variable X has pd f (1.1.576) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = βλβ xβ−1 , x > 0. ξ (x) q1 (x) − q2 (x)

Characterizations of Distributions

325

Corollary 2.156. The general solution of the differential equation in Corollary 2.155 is  Z  β β ξ (x) = e(λx) − βλβ xβ−1 e−(λx) (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.79. Let X : Ω
→ (−1, 1) be a continuous random variable and let q1 (x) ≡ 1 and q2 (x) = x + 1π sin(πx) , −1 < x < 1. The random variable X has pd f (1.1.578) if and only if the function ξ defined in Theorem 2.1 is of the form    1 1 1 + x + sin(πx) , − 1 < x < 1. ξ (x) = 2 π Corollary 2.157. Let X : Ω → (−1, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.79. The random variable X has pd f (1.1.578) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) 1 + cos (πx)
, = ξ (x) q1 (x) − q2 (x) 1 − x + π1 sin(πx)

− 1 < x < 1.

Corollary 2.158. The general solution of the differential equation in Corollary 2.157 is   −1  Z  1 −1 ξ (x) = 1 − x + sin(πx) − (1 + cos (πx)) (q1 (x)) q2 (x) dx + D , π where D is a constant. Proposition 2.80. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =
−1 and q2 (x) = q1 (x) e−θ/x , x > 0. The random variable X has pd f (1.1.590) if 1 + 2xθ2 and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

o 1n 1 + e−θ/x , x > 0. 2

Corollary 2.159. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.80. The random variable X has pd f (1.1.590) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) e−θ/x
, x > 0. = 2 ξ (x) q1 (x) − q2 (x) x 1 − e−θ/x

Corollary 2.160. The general solution of the differential equation in Corollary 2.159 is " Z #  −1 e−θ/x −1 −θ/x ξ (x) = 1 − e − (q1 (x)) q2 (x) dx + D , x2 where D is a constant.

G. G. Hamedani

326

Proposition 2.81. Let X : Ω → R be a continuous random variable and let q1 (x) = −1  ekx/(1+sgn(x)λ)φ ex and q2 (x) = q1 (x) ex , x ∈ R. The random variable X has pd f 2k (1+ekx/(1+sgn(x)λ)φ ) (1.1.596) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = ex , x ∈ R. 2 Corollary 2.161. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.81. The random variable X has pd f (1.1.596) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = −1, x ∈ R. ξ (x) q1 (x) − q2 (x) Corollary 2.162. The general solution of the differential equation in Corollary 2.161 is Z  1 x −1 −x ξ (x) = e e (q1 (x)) q2 (x) dx + D , 2 where D is a constant. Proposition 2.82. Let X : Ω → R be a continuous random variable and let q1 (x) = (2 − G (x))−1 and q2 (x) = q1 (x) e−G(x) , x ∈ R. The random variable X has pd f (1.1.598) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

 1  −G(x) e + e−1 , x ∈ R. 2

Corollary 2.163. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.82. The random variable X has pd f (1.1.598) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) g (x) e−G(x) , x ∈ R. = −G(x) ξ (x) q1 (x) − q2 (x) e + e−1 Corollary 2.164. The general solution of the differential equation in Corollary 2.163 is   −1  Z −1 −G(x) −1 −G(x) ξ (x) = e +e − g (x) e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.83. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  −1 β x β −δ( θx ) 1 − λ + 2λe and q2 (x) = q1 (x) e−δ( θ ) , x > 0. The random variable X has pd f (1.1.604) if and only if the function ξ defined in Theorem 2.1 is of the form x β 1 ξ (x) = e−δ( θ ) , x > 0. 2

Characterizations of Distributions

327

Corollary 2.165. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.83. The random variable X has pd f (1.1.604) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation δβ ξ0 (x) q1 (x) = β xβ−1 , x > 0. ξ (x) q1 (x) − q2 (x) θ Corollary 2.166. The general solution of the differential equation in Corollary 2.165 is  Z  β δβ β−1 −δ( θx )β −1 δ( θx ) ξ (x) = e − x e (q1 (x)) q2 (x) dx + D , θβ where D is a constant. Proposition 2.84. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = oα+1 n x µ −(x/σ)µ and q2 (x) = q1 (x) e−( σ ) , x > 0. The random variable X has pd f 1 + 1−e β (1.1.606) if and only if the function ξ defined in Theorem 2.1 is of the form x µ 1 ξ (x) = e−( σ ) , x > 0. 2

Corollary 2.167. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.84. The random variable X has pd f (1.1.606) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = µ (x/σ)µ−1 , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.168. The general solution of the differential equation in Corollary 2.167 is  Z  µ x µ µ−1 −( σx ) −1 ( ) σ ξ (x) = e − µ (x/σ) e (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.85. Let X : Ω → R be a continuous random variable and let q1 (x) =  −b+1 1 − (1 + G (x; η))−a and q2 (x) = q1 (x) (1 + G (x; η))−a , x ∈ R. The random variable X has pd f (1.1.620) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

 1 (1 + G (x; η))−a + 2−a , x ∈ R. 2

Corollary 2.169. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.85. The random variable X has pd f (1.1.620) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) ag (x; η) (1 + G (x; η))−a−1 = , x ∈ R. ξ (x) q1 (x) − q2 (x) (1 + G (x; η))−a − 2−a

G. G. Hamedani

328

Corollary 2.170. The general solution of the differential equation in Corollary 2.169 is  −1 ξ (x) = (1 + G (x; η))−a − 2−a ×  Z  −a−1 −1 − ag (x; η) (1 + G (x; η)) (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.86. Let X : Ω → (0, ∞) be a continuous random variable and let i n h  ioa+1 h  −θx− αβ (eβx−1) −θx− αβ (eβx −1) b − log 1 − 1 − e and q2 (x) = q1 (x) = 1 − 1 − e α βx −θx− β (e −1) q (x) e , x > 0. The random variable X has pd f (1.1.622) if and only if the 1

function ξ defined in Theorem 2.1 is of the form 1 −θx− αβ (eβx −1) ξ (x) = e , x > 0. 2 Corollary 2.171. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.86. The random variable X has pd f (1.1.622) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = θ + αeβx , x > 0. ξ (x) q1 (x) − q2 (x)

Corollary 2.172. The general solution of the differential equation in Corollary 2.171 is  Z    θx+ αβ (eβx−1) −θx− αβ (eβx −1) −1 βx ξ (x) = e − θ + αe e (q1 (x)) q2 (x) dx + D , where D is a constant.

Proposition 2.87. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = −1  α α e−(β/x) (1+λ−2λe−(β/x) ) and q2 (x) = q1 (x) x−α , x > 0. The random variable X 1−θ {1−e−(β/x)α [1+λ(1−e−(β/x)α )]} has pd f (1.1.624) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = x−α , x > 0. 2 Corollary 2.173. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.87. The random variable X has pd f (1.1.624) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = αx−1 , x > 0. ξ (x) q1 (x) − q2 (x)

Corollary 2.174. The general solution of the differential equation in Corollary 2.173 is  Z  ξ (x) = xα − αx−α−1 (q1 (x))−1 q2 (x) dx + D , where D is a constant.

Characterizations of Distributions

329

Proposition 2.88. Let h i X : Ω → R be a continuous random variable and let q1 (x) = G(x) 1 1 2 G(x) G(x) exp 2λ2 ln G(x) and q2 (x) = q1 (x) G(x) , x ∈ R. The random variable X has pd f (1.1.628) if and only if the function ξ defined in Theorem 2.1 is of the form   1 1 ξ (x) = + 1 , x ∈ R. 2 G (x) Corollary 2.175. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.88. The random variable X has pd f (1.1.628) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation g (x) (G (x))−2 ξ0 (x) q1 (x) = , x ∈ R. ξ (x) q1 (x) − q2 (x) (G (x))−1 − 1 Corollary 2.176. The general solution of the differential equation in Corollary 2.175 is  n o−1  Z −1 −2 −1 ξ (x) = (G (x)) − 1 − g (x) (G (x)) (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.89. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = −α (1 + λx−α ) eβx and q2 (x) = q1 (x) x−1 , x > 0. The random variable X has pd f (1.1.634) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

α −1 x , x > 0. α+1

Corollary 2.177. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.89. The random variable X has pd f (1.1.634) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = αx−1 , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.178. The general solution of the differential equation in Corollary 2.177 is  Z  ξ (x) = xα − αx−α−1 (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.90. Let X : Ω → R be a continuous random variable and let q1 (x) = −1 (1−2G(x)G(x) )  G(x)2  and q2 (x) = q1 (x) G (x)−1 , x ∈ R. The random variable X has pd f 2G(x)−1 φ

G(x)G(x)

(1.1.636) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

 1 G (x)−1 + 1 , x ∈ R. 2

G. G. Hamedani

330

Corollary 2.179. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.90. The random variable X has pd f (1.1.636) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) g (x) G (x)−2 = , x ∈ R. ξ (x) q1 (x) − q2 (x) G (x)−1 − 1

Corollary 2.180. The general solution of the differential equation in Corollary 2.179 is  h i−1  Z −1 −2 −1 ξ (x) = G (x) − 1 − g (x) G (x) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.91. Let X : Ω → R be a continuous random variable and let q1 (x) = n α
α o−1 1+θG(x;φ) −θ(− log G(x;φ) ) , x ∈ R. The random 1 + − logG (x; φ) and q (x) = q (x) e 2 1 G(x;φ) variable X has pd f (1.1.638) if and only if the function ξ defined in Theorem 2.1 is of the form α 1 ξ (x) = e−θ(− log G(x;φ)) , x ∈ R. 2

Corollary 2.181. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.91. The random variable X has pd f (1.1.638) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation
α−1 αθg (x; φ) − log G(x; φ) ξ0 (x) q1 (x) , x ∈ R. = ξ (x) q1 (x) − q2 (x) G (x; φ)

Corollary 2.182. The general solution of the differential equation in Corollary 2.181 is ξ (x) = eθ(− logG(x;φ)) × " Z #
α−1 αθg (x; φ) − logG (x; φ) − e−θ(− log G(x;φ)) (q1 (x))−1 q2 (x) dx + D , G (x; φ) where D is a constant. Proposition 2.92. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =   a  a 1−α − x − x 1−e ( b ) − βc ln 1−e ( b ) a −( bx ) , x > 0. The random   α    α θ−1 and q2 (x) = q1 (x) e a a −( x ) −( x ) b b   −− c ln1−e −− c ln1−e e

β

 1−e 

β

  

variable X has pd f (1.1.646) if and only if the function ξ defined in Theorem 2.1 is of the form x a 1 ξ (x) = e−( b ) , x > 0. 2

Characterizations of Distributions

331

Corollary 2.183. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.92. The random variable X has pd f (1.1.646) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation a  x a−1 ξ0 (x) q1 (x) = , x > 0. ξ (x) q1 (x) − q2 (x) b b

Corollary 2.184. The general solution of the differential equation in Corollary 2.183 is  Z    x a a x a−1 −( bx )a −1 ( ) b ξ (x) = e − e (q1 (x)) q2 (x) dx + D , b b where D is a constant.

Proposition 2.93. Let X : Ω → R be a continuous random variable and let q1 (x) = o n θ 1−α θ  1+λ(1−G(x;φ)2 ) 2  α G (x; φ) , x ∈ R. The random variable X and q (x) = q (x) 1 − 2 1 θ 1− 1+λ(1−G(x;φ)2 ) e has pd f (1.1.648) if and only if the function ξ defined in Theorem 2.1 is of the form   θ  1 2 ξ (x) = 1 + 1 − G (x; φ) , x ∈ R. 2

Corollary 2.185. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.93. The random variable X has pd f (1.1.648) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q

1 (x)

ξ (x) q1 (x) − q2 (x)

=

 θ−1 1 − G (x; φ)2 , x ∈ R.  θ 2 1 − 1 − G (x; φ)

θ 2 g (x; φ)G (x; φ)

Corollary 2.186. The general solution of the differential equation in Corollary 2.185 is   θ −1 2 ξ (x) = 1 − 1 − G (x; φ) ×  Z   θ−1 θ 2 −1 − g (x; φ)G (x; φ) 1 − G (x; φ) (q1 (x)) q2 (x) dx + D , 2 where D is a constant. Proposition 2.94.Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = h iβ  −2θb a (1+x ) (1 + xa )θb − 1 and q2 (x) = q1 (x) (1 + xa )−θb , x > 0. The random β−1 exp [(1+xa )θb −1] variable X has pd f (1.1.650) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 (1 + xa )−θb , x > 0. 2

G. G. Hamedani

332

Corollary 2.187. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.94. The random variable X has pd f (1.1.650) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation abθxa−1 ξ0 (x) q1 (x) = , x > 0. ξ (x) q1 (x) − q2 (x) 1 + xa Corollary 2.188. The general solution of the differential equation in Corollary 2.187 is  Z  −1 a θb a−1 a −θb ξ (x) = (1 + x ) − abθx (1 + x ) (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.95. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = h  i −1 β β 1−a λxβ e−λx (1+xβ) e−λx 1+ 1+λ β and q2 (x) = q1 (x) e−λx , x > 0. The random variable X has n h  i ob−1 β β a 1−

λx 1+ 1+λ e−λx

pd f (1.1.658) if and only if the function ξ defined in Theorem 2.1 is of the form β 1 ξ (x) = e−λx , x > 0. 2

Corollary 2.189. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.95. The random variable X has pd f (1.1.658) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = λβxβ−1 , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.190. The general solution of the differential equation in Corollary 2.189 is  Z  β β ξ (x) = eλx − λβxβ−1 e−λx (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.96. Let X : Ωo→ R be a continuous random variable and let q1 (x) = n 2 2α  G (x; φ) + 1 − G (x; φ)α and q2 (x) = q1 (x) G (x; φ)α , x ∈ R. The random variable X has pd f (1.1.328) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 1 + G (x; φ)α , x ∈ R. 2

Corollary 2.191. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.96. The random variable X has pd f (1.1.328) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) αg (x; φ)G (x; φ)α−1 = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; φ)α

Characterizations of Distributions

333

Corollary 2.192. The general solution of the differential equation in Corollary 2.191 is  Z   −1 ξ (x) = 1 − G (x; φ)α − αg (x; φ)G (x; φ)α−1 (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

Proposition 2.97. Let X : Ω → R be a continuous random variable and let q1 (x) =  1−b and q2 (x) = q1 (x) G (x; ϕ)α , x ∈ R. The random variable X has pd f 1 − αG (x; ϕ)α (1.1.662) if and only if the function ξ defined in Theorem 2.1 is of the form 1 1 + G (x; ϕ)α , x ∈ R. 2 Corollary 2.193. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.97. The random variable X has pd f (1.1.662) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ (x) =

ξ0 (x) q1 (x) αg (x; ϕ) G (x; ϕ)α−1 = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; ϕ)α

Corollary 2.194. The general solution of the differential equation in Corollary 2.193 is  Z   α −1 α−1 −1 ξ (x) = 1 − G (x; ϕ) − αg (x; ϕ) G (x; ϕ) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.98. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = i2 h
1/α
1/α + (1 − δ) and q2 (x) = q1 (x) 1 + αβαθ x−αθ , x > 0. The ranδ 1 + αβαθ x−αθ dom variable X has pd f (1.1.670) if and only if the function ξ defined in Theorem 2.1 is of the form   1/α 1  αθ −αθ ξ (x) = 1 + αβ x + 1 , x > 0. 2

Corollary 2.195. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.98. The random variable X has pd f (1.1.670) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation
1 −1 αβαθ x−αθ−1 1 + αβαθ x−αθ α ξ0 (x) q1 (x) = , x > 0. 1/α ξ (x) q1 (x) − q2 (x) (1 + αβαθ x−αθ ) − 1

Corollary 2.196. The general solution of the differential equation in Corollary 2.195 is  −1 1/α αθ −αθ ξ (x) = 1 + αβ x −1 ×  Z    1 −1 −1 αθ −αθ−1 αθ −αθ α − αβ x 1 + αβ x (q1 (x)) q2 (x) dx + D ,

where D is a constant.

G. G. Hamedani

334

Proposition 2.99. oLet X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = n −1/σ
−1/σ exp B(eθx −1) θx , x > 0. The random variable X h n oiα−1 and q2 (x) = q1 (x) e − 1 −1/σ 1−exp −B(eθx −1) has pd f (1.1.334) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

−1/σ 1  θx e −1 , x > 0. 2

Corollary 2.197. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.99. The random variable X has pd f (1.1.334) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation −1 θ  ξ0 (x) q1 (x) = eθx eθx − 1 , x > 0. ξ (x) q1 (x) − q2 (x) σ

Corollary 2.198. The general solution of the differential equation in Corollary 2.197 is  − 1 −1  1/σ  Z θ  σ −1 θx θx θx e e −1 ξ (x) = e − 1 − (q1 (x)) q2 (x) dx + D , σ where D is a constant.

Proposition 2.100. Let X : Ω → R be a continuous random variable and let q1 (x) = n  2 o−1 θ + 21 θ2 G (x; η) and q2 (x) = q1 (x) G (x; η) , x ∈ R. The random variable X has pd f (1.1.676) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

θ G (x; η) , x ∈ R. 1+θ

Corollary 2.199. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.100. The random variable X has pd f (1.1.676) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) θg (x; η) = , x ∈ R. ξ (x) q1 (x) − q2 (x) G(x; η)

Corollary 2.200. The general solution of the differential equation in Corollary 2.199 is  Z  ξ (x) = G (x; η)−1 − θg (x; η) (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.101. Let X : Ω → R be a continuous random variable and let q1 (x) = h    √ i−1
−1 2 x 1 + π arctan √ax3 + 3+a3ax and q2 (x) = q1 (x) 3 + x2 , x ∈ R. The random 2 x2 variable X has pd f (1.1.694) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =


−1 1 3 + x2 , x ∈ R. 2

Characterizations of Distributions

335

Corollary 2.201. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.101. The random variable X has pd f (1.1.694) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation
−1 ξ0 (x) q1 (x) = x 3 + x2 , x ∈ R. ξ (x) q1 (x) − q2 (x)

Corollary 2.202. The general solution of the differential equation in Corollary 2.201 is  Z 
−1 2 ξ (x) = 3 + x − x (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.102.Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  β β λ λ 1−( λ+x ) and q (x) = q (x) α1−( λ+x ) , x > 0. The random variable X has pd f 1+α+α 2 1 (1.1.696) if and only if the function ξ defined in Theorem 2.1 is of the form   β λ 1 1−( λ+x ) α+α , x > 0. ξ (x) = 2

Corollary 2.203. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.102. The random variable X has pd f (1.1.696) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation λ

β

log(α) βλβ (λ + x)−(β+1) α1−( λ+x ) ξ0 (x) q1 (x) = , x > 0. β λ ξ (x) q1 (x) − q2 (x) α − α1−( λ+x )

Corollary 2.204. The general solution of the differential equation in Corollary 2.203 is   β −1 λ 1−( λ+x ) ξ (x) = α − α ×  Z  λ β − log(α) βλβ (λ + x)−(β+1) α1−( λ+x ) (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.103. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = h

xe

θ γ

 iγ+1 1+ βx e−x/β

  −γ ) 1− 1+ x e−x/β β

(

and q2 (x) = q1 (x) e−x/β , x > 0.

The random variable X has pd f

(1.1.700) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−x/β , x > 0. 2

G. G. Hamedani

336

Corollary 2.205. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.103. The random variable X has pd f (1.1.700) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation 1 ξ0 (x) q1 (x) = , x > 0. ξ (x) q1 (x) − q2 (x) β

Corollary 2.206. The general solution of the differential equation in Corollary 2.205 is  Z  1 −x/β −1 x/β e (q1 (x)) q2 (x) dx + D , ξ (x) = e − β where D is a constant. Proposition 2.104. Let X : Ω → R be a continuous random variable and let q1 (x) = n o−1 1−[1−G(x;η)]β n oα−1 2−[1−G(x;η)]β

and q2 (x) = q1 (x) [1 − G (x; η)] , x ∈ R. The random variable X has

pd f (1.1.704) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

αβ [1 − G (x; η)] , x ∈ R. αβ + 1

Corollary 2.207. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.104. The random variable X has pd f (1.1.704) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) αβg (x; η) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; η)

Corollary 2.208. The general solution of the differential equation in Corollary 2.207 is ξ (x) = {[1 − G (x; η)]}−αβ ×  Z  αβ−1 −1 − αβg (x; η) [1 − G (x; η)] (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.105. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = Rx 1 and q (x) = q (x) G (x) , x > 0, G (x) = g (u)du. The random variable X has 2 1 0 M(θH(x)) pd f (1.1.730) if and only if the function ξ defined in Theorem 2.1 is of the form 1 [1 + G (x)] , x > 0. 2 Corollary 2.209. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.105. The random variable X has pd f (1.1.730) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ (x) =

ξ0 (x) q1 (x) g (x) = , x > 0. ξ (x) q1 (x) − q2 (x) 1 − G (x)

Characterizations of Distributions

337

Corollary 2.210. The general solution of the differential equation in Corollary 2.209 is  Z  ξ (x) = [1 − G (x)]−1 − g (x) (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.106. Let X : Ω → R be a continuous random variable and let q1 (x) = i2 h Rx 2+α2 and q2 (x) = q1 (x) Φ (x) , x ∈ R, Φ (x) = −∞ φ (u) du. The random variable 2 (1−αx) +1

X has pd f (1.1.734) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 [1 + Φ (x)], x ∈ R. 2

Corollary 2.211. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.106. The random variable X has pd f (1.1.734) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) φ (x) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − Φ (x) Corollary 2.212. The general solution of the differential equation in Corollary 2.211 is  Z  −1 −1 ξ (x) = [1 − Φ (x)] − φ (x) (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.107. Let X : Ω → R be a continuous random variable and let q1 (x) = h i−1
2 −1 − 1 − αx − βx3 + 1 and q2 (x) = q1 (x) (1 − e−x ) , x ∈ R. The random variable X has pd f (1.1.736) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =


−1 1 1 − e−x , x ∈ R. 2

Corollary 2.213. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.107. The random variable X has pd f (1.1.736) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) e−x = , x ∈ R. ξ (x) q1 (x) − q2 (x) (1 − e−x ) Corollary 2.214. The general solution of the differential equation in Corollary 2.213 is  Z 
−1 −x −x ξ (x) = 1 − e − e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

G. G. Hamedani

338

Proposition 2.108. Let X : Ω → (A, B) be a continuous random variable and let q1 (x) ≡ 1 −x and q2 (x) = e−e , A < x < B. The random variable X has pd f (1.1.742) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

 1 −x 1 + e−e , A < x < B. 2

Corollary 2.215. Let X : Ω → (A, B) be a continuous random variable and let q1 (x) be as in Proposition 2.108. The random variable X has pd f (1.1.742) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation −x

e−e e−x ξ0 (x) q1 (x) = , A < x < B. ξ (x) q1 (x) − q2 (x) 1 − e−e−x

Corollary 2.216. The general solution of the differential equation in Corollary 2.215 is   −1  Z −1 −e−x −e−x −x ξ (x) = 1 − e − e e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.109. Let X : Ω → R be a continuous random variable and let q1 (x) = n oθ G(x;ζ) {1−G(x;ζ)[1+G(x;ζ)]/2}θ+1 λ 1−G(x;ζ)[1+G(x;ζ)]/2 e [1+G(x;ζ)2 /2]

and q2 (x) = q1 (x) G (x; ζ)θ , x ∈ R. The random

variable X has pd f (1.1.756) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

 1 1 + G (x; ζ)θ , x ∈ R. 2

Corollary 2.217. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.109. The random variable X has pd f (1.1.756) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) θg (x; ζ) G (x; ζ)θ−1 = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; ζ)θ Corollary 2.218. The general solution of the differential equation in Corollary 2.217 is   −1  Z θ θ−1 −1 ξ (x) = 1 − G (x; ζ) − θg (x; ζ) G (x; ζ) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.110. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) = [x (1 − α) + α]θ+1 and q2 (x) = q1 (x) x, 0 < x < 1. The random variable X has pd f (1.1.758) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 (1 + x) , 0 < x < 1. 2

Characterizations of Distributions

339

Corollary 2.219. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.110. The random variable X has pd f (1.1.758) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation 1 ξ0 (x) q1 (x) = , 0 < x < 1. ξ (x) q1 (x) − q2 (x) 1 − x

Corollary 2.220. The general solution of the differential equation in Corollary 2.219 is  Z  −1 −1 ξ (x) = (1 − x) − (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.111. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = (1+γx−β)

λζα γ +1

ζα +1 1+γx−β γ

1−(

)

!η−1

and q2 (x) = q1 (x) x−1 , x > 0.

The random variable X has pd f

(1.1.760) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

β −1 x , x > 0. β+1

Corollary 2.221. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.111. The random variable X has pd f (1.1.760) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = βx−1 , x > 0. ξ (x) q1 (x) − q2 (x)

Corollary 2.222. The general solution of the differential equation in Corollary 2.221 is  Z  −1 β −β−1 ξ (x) = x − βx (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.112. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  β+1 (1+a+ax)2 e−2ax (1+a)eax α − and q2 (x) = q1 (x) e−ax , x > 0. The random variable X α (1+x) α(1+a+ax) has pd f (1.1.762) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−ax , x > 0. 2 Corollary 2.223. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.112. The random variable X has pd f (1.1.762) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = a, x > 0. ξ (x) q1 (x) − q2 (x)

G. G. Hamedani

340

Corollary 2.224. The general solution of the differential equation in Corollary 2.223 is  Z  −1 ax −ax ξ (x) = e − ae (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.113. Let X : Ω → (0, a continuous random variable and let q1 (x) = h∞) be iβ   −2ax ax ax e (1+a+ax)[(1+a)e −α(1+a+ax)] (1+a)e − αα and q2 (x) = q1 (x) e−ax , x > exp log α(1+a+ax) h  iβ−1 (1+a)eax (1+x) log

α α(1+a+ax) − α

0. The random variable X has pd f (1.1.764) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−ax , x > 0. 2 Corollary 2.225. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.113. The random variable X has pd f (1.1.764) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = a, x > 0. ξ (x) q1 (x) − q2 (x)

Corollary 2.226. The general solution of the differential equation in Corollary 2.225 is  Z  −1 ax −ax ξ (x) = e − ae (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.114. Let X : Ω → (0, ∞)be a continuous random variable and let q1 (x) = h i  θ 1 + cG (1/x)β 1 + cG (1/x)β − 1 and q2 (x) = q1 (x) G (1/x), x > 0. The random variable X has pd f (1.1.766) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

β G (1/x) , x > 0. β+1

Corollary 2.227. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.114. The random variable X has pd f (1.1.766) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) βx−2 g (1/x) = , x > 0. ξ (x) q1 (x) − q2 (x) G (1/x)

Corollary 2.228. The general solution of the differential equation in Corollary 2.227 is  Z  −β −1 −2 ξ (x) = G (1/x) − βx g (1/x) (q1 (x)) q2 (x) dx + D , where D is a constant.

Characterizations of Distributions

341

Proposition 2.115. Let X : Ω → (a, b) be a continuous random variable and let q1 (x) = h
−α
α i1−β
α ln ax and q2 (x) = q1 (x) ln ax , x > 0. The random variable X has 1 − ln ab pd f (1.1.770) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

 x α o 1n 1 − ln , a < x < b. 2 a

Corollary 2.229. Let X : Ω → (a, b) be a continuous random variable and let q1 (x) be as in Proposition 2.115. The random variable X has pd f (1.1.770) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation
α−1 αx−1 ln ax ξ0 (x) q1 (x) =
α , 0 < x < b. ξ (x) q1 (x) − q2 (x) 1 − ln ax

Corollary 2.230. The general solution of the differential equation in Corollary 2.229 is  n  x α o−1  Z  x α−1 −1 −1 ξ (x) = 1 − ln − αx ln (q1 (x)) q2 (x) dx + D , a a where D is a constant.

Proposition 2.116. Let X : Ω → (β, ∞) be a continuous random variable and let q1 (x) = 1/ logx and q2 (x) = q1 (x) x−α , x > β. The random variable X has pd f (1.1.772) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = x−α , x > β. 2 Corollary 2.231. Let X : Ω → (β, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.116. The random variable X has pd f (1.1.772) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = αx−1 , x > β. ξ (x) q1 (x) − q2 (x) Corollary 2.232. The general solution of the differential equation in Corollary 2.231 is  Z  −1 α −α−1 ξ (x) = x − αx (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.117. Let X
: Ω → R be a continuous random variable and let q1 (x) =
[ 1 − ρθG (x) 1 − θG (x) ]2 and q2 (x) = q1 (x) [1 − ρθ2 G(x)]2 , x ∈ R. The random variable X has pd f (1.1.778) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 1 + [1 − ρθ2 G (x)]2 , x ∈ R. 2

G. G. Hamedani

342

Corollary 2.233. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.117. The random variable X has pd f (1.1.778) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation 2ρθ2 g (x) [1 − ρθ2 G (x)] ξ0 (x) q1 (x) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − [1 − ρθ2 G (x)]2

Corollary 2.234. The general solution of the differential equation in Corollary 2.233 is −1  × ξ (x) = 1 − [1 − ρθ2 G (x)]2  Z  −1 2 2 − 2ρθ g (x) [1 − ρθ G (x)] (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.118. Let X : Ω → R be a continuous random variable and let q1 (x) = [(1−ρθG(x))(1−θG(x))]r+1 h i r+1 and q2 (x) = q1 (x) G (x) , x ∈ R. The random variable X has 1+ρ−2ρθG(x)−ρ(1−θG(x)) pd f (1.1.780) if and only if the function ξ defined in Theorem 2.1 is of the form 1 {1 + G (x)}, x ∈ R. 2 Corollary 2.235. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.118. The random variable X has pd f (1.1.780) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ (x) =

ξ0 (x) q1 (x) g (x) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x)

Corollary 2.236. The general solution of the differential equation in Corollary 2.235 is  Z  −1 −1 ξ (x) = {1 − G (x)} − g (x) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.119. Let X : Ω → R be a continuous random variable and let q1 (x) = n o−1
m−1  
m θGx + 1 2ρθG(x) + ρ + 1 − ρ and q2 (x) = q1 (x) ρθGx + 1 , x ∈ R. The random variable X has pd f (1.1.782) if and only if the function ξ defined in Theorem 2.1 is of the form
m 1 1 + ρθG(x) + 1 , x ∈ R. 2 Corollary 2.237. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.119. The random variable X has pd f (1.1.782) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ (x) =


m−1 mρθg (x) ρθG (x) + 1 ξ0 (x) q1 (x) =
m , x ∈ R. ξ (x) q1 (x) − q2 (x) ρθG (x) + 1 − 1

Characterizations of Distributions

343

Corollary 2.238. The general solution of the differential equation in Corollary 2.237 is 
m −1 ξ (x) = ρθG (x) + 1 − 1 ×  Z 
m−1 −1 − mρθg (x) ρθG(x) + 1 (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.120. Let X : Ω → R be a continuous random variable and let q1 (x) = n h io−1 eλρG(x) (1 + ρ) eλG(x) − ρ and q2 (x) = q1 (x) G (x) , x ∈ R. The random variable X has pd f (1.1.784) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 {1 + G (x)}, x ∈ R. 2

Corollary 2.239. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.120. The random variable X has pd f (1.1.784) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation g (x) ξ0 (x) q1 (x) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x)

Corollary 2.240. The general solution of the differential equation in Corollary 2.239 is  Z  −1 −1 ξ (x) = {1 − G (x)} − g (x) (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.121. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = γ exp{−2(x/β)γ +(e(x/β) −1)} −(x/β)γ and q (x) = q (x) e , x > 0. The random variable X has pd f γ 2 1 (x/β) −1) (e (1.1.796) if and only if the function ξ defined in Theorem 2.1 is of the form γ 1 ξ (x) = e−(x/β) , x > 0. 2

Corollary 2.241. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.121. The random variable X has pd f (1.1.796) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) γ = γ xγ−1 , x > 0. ξ (x) q1 (x) − q2 (x) β

Corollary 2.242. The general solution of the differential equation in Corollary 2.241 is  Z  γ γ−1 −(x/β)γ −1 (x/β)γ ξ (x) = e − x e (q1 (x)) q2 (x) dx + D , βγ where D is a constant.

G. G. Hamedani

344

Proposition 2.122. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  −λ/x −(β−1) (   ) αe −1 α−1



−λ/x α−αe α−1

−(β+1)

exp ϕ

−λ/x

αe −1 −λ/x α−αe

β

and q2 (x) = q1 (x) αe

−λ/x

, x > 0. The random variable

X has pd f (1.1.800) if and only if the function ξ defined in Theorem 2.1 is of the form  1 e−λ/x ξ (x) = α+α , x > 0. 2 Corollary 2.243. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.122. The random variable X has pd f (1.1.800) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation −λ/x

ξ0 (x) q1 (x) λ ln(α)e−λ/xαe
, x > 0. = −λ/x ξ (x) q1 (x) − q2 (x) x2 α − αe

Corollary 2.244. The general solution of the differential equation in Corollary 2.243 is # " Z −λ/x  −1 λ ln(α)e−λ/xαe −1 e−λ/x (q1 (x)) q2 (x) dx + D , ξ (x) = α − α − x2 where D is a constant.

Proposition 2.123. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = h i1−α

d+x −λd/2 d+x −1 1 − d−x and q2 (x) = q1 (x) d−x , 0 < x < d. The random variable X has pd f (1.1.802) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

λd λd + 2



d +x d −x

−1

, 0 < x < d.

Corollary 2.245. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.123. The random variable X has pd f (1.1.802) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) λd 2 = 2 , 0 < x < d. ξ (x) q1 (x) − q2 (x) d − x2

Corollary 2.246. The general solution of the differential equation in Corollary 2.245 is #  " Z d +x λd 2 ξ (x) = − (q1 (x))−1 q2 (x) dx + D , d −x (d + x)2 where D is a constant. Proposition 2.124. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  −(b+1) −c/x α−1 and q2 (x) = q1 (x) αe , x > 0. The random variable X has pd f −c/x e α−α

(1.1.804) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

i 1h −c/x α + αe , x > 0. 2

Characterizations of Distributions

345

Corollary 2.247. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.124. The random variable X has pd f (1.1.804) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation −c/x

c log α e−c/xαe ξ0 (x) q1 (x)
, x > 0. = −c/x ξ (x) q1 (x) − q2 (x) x2 α − αe

Corollary 2.248. The general solution of the differential equation in Corollary 2.247 is   −1  Z c log α −1 e−c/x −c/x e−c/x ξ (x) = α − α − e α (q1 (x)) q2 (x) dx + D , x2 where D is a constant.

Proposition 2.125. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = (λ+1) [βγ −(β−e−x )γ ] −x γ ( "  λ #) and q2 (x) = q1 (x) (β − e ) , x > 0. The random variable X has exp

θ λ

1−

βγ −(β−1)γ γ βγ −(β−e−x )

pd f (1.1.806) if and only if the function ξ defined in Theorem 2.1 is of the form


γ i 1h γ β + β − e−x , x > 0. 2 Corollary 2.249. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.125. The random variable X has pd f (1.1.806) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ (x) =

γ−1

ξ0 (x) q1 (x) γe−x (β − e−x ) = γ , x > 0. ξ (x) q1 (x) − q2 (x) β − (β − e−x )γ

Corollary 2.250. The general solution of the differential equation in Corollary 2.249 is  Z  
γ−1
γ−1 ξ (x) = βγ + β − e−x − γe−x β − e−x (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

Proposition 2.126. Let X : Ω → R be a continuous random variable and let q1 (x) = n h ioβ  α G(x;η)[G(x;η)α +G(x;η)α ] G(x;η) and q2 (x) = q1 (x) G (x; η) , x ∈ − log G(x;η) n h ioβ−1 exp α +G(x;η)α G(x;η)α − log

G(x;η)α +G(x;η)α

R. The random variable X has pd f (1.1.808) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

α G (x; η) , x ∈ R. α+1

Corollary 2.251. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.126. The random variable X has pd f (1.1.808) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) αg (x; η) = , x ∈ R. ξ (x) q1 (x) − q2 (x) G(x; η)

G. G. Hamedani

346

Corollary 2.252. The general solution of the differential equation in Corollary 2.251 is  Z  ξ (x) = G(x; η)−1 − αg (x; η) (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.127. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =   −1
−1 2 G x2 ; p+3 , σ and q2 (x) = q1 (x) σx , x > 0. The random variable X has pd f 2 (1.1.812) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

p  x −1 , x > 0. p+1 σ

Corollary 2.253. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.127. The random variable X has pd f (1.1.812) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = px−1 , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.254. The general solution of the differential equation in Corollary 2.253 is   x  Z  −1 −1 −1 x ξ (x) = − px (q1 (x)) q2 (x) dx + D , σ σ where D is a constant. Proposition 2.128. Let X : Ω → (m, g) be a continuous random variable and let q1 (x) = h i−1 −β −β 1 + λ − 2λe−αx and q2 (x) = q1 (x) e−αx , m < x < g. The random variable X has pd f (1.1.816) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

 1 −β 1 + e−αx , m < x < g. 2

Corollary 2.255. Let X : Ω → (m, g) be a continuous random variable and let q1 (x) be as in Proposition 2.128. The random variable X has pd f (1.1.816) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation −β

ξ0 (x) q1 (x) αβx−β−1 e−αx = , m < x < g. −β ξ (x) q1 (x) − q2 (x) 1 − e−αx

Corollary 2.256. The general solution of the differential equation in Corollary 2.255 is  −1  Z  −1 −β−1 −αx−β −αx−β ξ (x) = 1 − e − αβx e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Characterizations of Distributions

347

Proposition 2.129. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = 2 (1−log{1−[1−(1+xα1 )−α2 ]}) and q2 (x) = q1 (x) (1 + xα1 )−1 , x > 0. The random variable X −α α [1−log{1−[1−(1+x 1 ) 2 ]}]+1 has pd f (1.1.818) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

α2 −1 (1 + xα1 ) , x > 0. α2 + 1

Corollary 2.257. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.129. The random variable X has pd f (1.1.818) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) α1 α2 xα1 −1 = , x > 0. ξ (x) q1 (x) − q2 (x) 1 + xα1 Corollary 2.258. The general solution of the differential equation in Corollary 2.257 is  Z  −1 α1 −1 α1 −1 ξ (x) = (1 + x ) − α1 α2 x (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.130. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = h i−2 (1 − α log x)2 + 1 and q2 (x) = q1 (x) Φ (logx) , x > 0. The random variable X has pd f (1.1.828) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = Φ (logx) , x > 0. 2 Corollary 2.259. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.130. The random variable X has pd f (1.1.828) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) φ (logx) =− , x > 0. ξ (x) q1 (x) − q2 (x) xΦ (logx) Corollary 2.260. The general solution of the differential equation in Corollary 2.259 is Z  φ (logx) −1 −1 ξ (x) = (Φ (logx)) (q1 (x)) q2 (x) dx + D , x where D is a constant. Proposition 2.131. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) ≡ 1 b d g and q2 (x) = q1 (x) e−(ax +cx +ex ) , x > 0. The random variable X has pd f (1.1.830) if and only if the function ξ defined in Theorem 2.1 is of the form b d g 1 ξ (x) = e−(ax +cx +ex ) , x > 0. 2

G. G. Hamedani

348

Corollary 2.261. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.131. The random variable X has pd f (1.1.830) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = abxb−1 + cdxd−1 + egxg−1 , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.262. The general solution of the differential equation in Corollary 2.261 is b d g ξ (x) = e(ax +cx +ex )× Z    −1 b−1 d−1 g−1 −(axb +cxd +exg ) abx + cdx + egx e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.132. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = x1−θ −θx , x > 0. The random variable X has pd f (1.1.842) if and 1+θ+θx and q2 (x) = q1 (x) e only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−θx , x > 0. 2 Corollary 2.263. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.132. The random variable X has pd f (1.1.842) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = θ, x > 0. ξ (x) q1 (x) − q2 (x)

Corollary 2.264. The general solution of the differential equation in Corollary 2.263 is  Z  −1 θx −θx ξ (x) = e − θe (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.133. Let X : Ω → R be a continuous random variable and let q1 (x) ≡ 1 and q2 (x) = (1 + G (x; η))−α , x ∈ R. The random variable X has pd f (1.1.856) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 (1 + G (x; η))−α + 2−α , x ∈ R. 2

Corollary 2.265. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.133. The random variable X has pd f (1.1.856) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) αg (x; η) (1 + G (x; η))−α−1 = , x ∈ R. ξ (x) q1 (x) − q2 (x) (1 + G (x; η))−α + 2−α

Characterizations of Distributions

349

Corollary 2.266. The general solution of the differential equation in Corollary 2.265 is  Z   −1 ξ (x) = (1 + G (x; η))−α + 2−α − αg (x; η) (1 + G (x; η))−α−1 q2 (x) dx + D ,

where D is a constant.

Proposition 2.134. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = n h io2 α α e−βx 1−θ β+1+βx β+1

α

and q2 (x) = q1 (x) e−βx , x > 0. The random variable X has pd f (1.1.858) if and only if the function ξ defined in Theorem 2.1 is of the form 1+xα

1 α ξ (x) = e−βx , x > 0. 2 Corollary 2.267. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.134. The random variable X has pd f (1.1.858) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = αβxα−1 , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.268. The general solution of the differential equation in Corollary 2.267 is  Z  −1 βxα α−1 −βxα ξ (x) = e − αβx e (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.135. Let X : Ω → (0, 2π) be a continuous random variable and let q1 (x) =  −1 1+x 2πe−2πλ and q2 (x) = q1 (x) e−λx , 0 < x < 2π. The random variable X has + 2 1−e−2πλ (1−e−2πλ) pd f (1.1.876) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−λx , 0 < x < 2π. 2 Corollary 2.269. Let X : Ω → (0, 2π) be a continuous random variable and let q1 (x) be as in Proposition 2.135. The random variable X has pd f (1.1.876) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = λ, 0 < x < 2π. ξ (x) q1 (x) − q2 (x) Corollary 2.270. The general solution of the differential equation in Corollary 2.269 is  Z  −1 λx −λx ξ (x) = e − λe (q1 (x)) q2 (x) dx + D , where D is a constant.

G. G. Hamedani

350

Proposition 2.136. Let X : Ω → R be a continuous random variable and let q1 (x) =  −1 1 + G (x; ζ)2α and q2 (x) = q1 (x) G (x; ζ)α , x ∈ R. The random variable X has pd f (1.1.878) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =


1 1 + G (x; ζ)α , x ∈ R. 2

Corollary 2.271. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.136. The random variable X has pd f (1.1.878) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) αg (x; ζ)α G (x; ζ)α−1 = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; ζ)α Corollary 2.272. The general solution of the differential equation in Corollary 2.271 is  Z  α
−1 α α−1 −1 ξ (x) = 1 − G (x; ζ) − αg (x; ζ) G (x; ζ) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.137. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = x−α−1    and q2 (x) = q1 (x) x−1 , x > 0. The random variable X    1/q α  R 1/q α 1 α/q 0 t

1+

xt

β

exp −

xt

β

dt

has pd f (1.1.880) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = x−1 , x > 0. 2

Corollary 2.273. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.137. The random variable X has pd f (1.1.880) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = x−1 , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.274. The general solution of the differential equation in Corollary 2.273 is  Z  −1 −2 ξ (x) = x − x (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.138. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = ρ(x)1−α 1+βx

and q2 (x) = q1 (x) e−αx , x > 0. The random variable X has pd f (1.1.882) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−αx , x > 0. 2

Characterizations of Distributions

351

Corollary 2.275. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.138. The random variable X has pd f (1.1.882) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = α, x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.276. The general solution of the differential equation in Corollary 2.275 is  Z  −1 αx −αx ξ (x) = e − αe (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.139. Let X : Ω → R be a continuous random variable and let q1 (x) =  
−x a 1−b −e−x 1 − {1 − exp [−e ]} and q2 (x) = q1 (x) 1 − e , x > 0. The random variable X has pd f (1.1.884) if and only if the function ξ defined in Theorem 2.1 is of the form  a  −x ξ (x) = 1 − e−e , x ∈ R. a+1 Corollary 2.277. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.139. The random variable X has pd f (1.1.884) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation −x

ξ0 (x) q1 (x) ae−x−e = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − e−e−x

Corollary 2.278. The general solution of the differential equation in Corollary 2.277 is   −1  Z −1 −e−x −x−e−x (q1 (x)) q2 (x) dx + D , ξ (x) = 1 − e − ae

where D is a constant.

Proposition 2.140. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) = θx (1 − x) and q2 (x) = q1 (x) e− 1−x , 0 < x < 1. The random variable X has pd f (1.1.886) if and only if the function ξ defined in Theorem 2.1 is of the form 1 θx ξ (x) = e− 1−x , 0 < x < 1. 2 Corollary 2.279. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.140. The random variable X has pd f (1.1.886) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = θ (1 − x)−2 , 0 < x < 1. ξ (x) q1 (x) − q2 (x)

G. G. Hamedani

352

Corollary 2.280. The general solution of the differential equation in Corollary 2.279 is  Z  θx θx −2 − 1−x −1 1−x − θ (1 − x) e (q1 (x)) q2 (x) dx + D , ξ (x) = e where D is a constant. Proposition 2.141. h i Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) = β exp α (− logx) and q2 (x) = q1 (x) (− log x) , 0 < x < 1. The random variable X has pd f (1.1.894) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

β (− logx) , 0 < x < 1. β+1

Corollary 2.281. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.141. The random variable X has pd f (1.1.894) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation β ξ0 (x) q1 (x) =− , 0 < x < 1. ξ (x) q1 (x) − q2 (x) x log x

Corollary 2.282. The general solution of the differential equation in Corollary 2.281 is Z  β −1 −1 ξ (x) = (log x) (q1 (x)) q2 (x) dx + D , x where D is a constant. Proposition 2.142. Let
 X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = 
α  α θx θx −θx −θx and q2 (x) = q1 (x) 1 − e 1 + 1+θ , x > 0. The random exp −1 + e 1 + 1+θ variable X has pd f (1.1.896) if and only if the function ξ defined in Theorem 2.1 is of the form    α  1 θx ξ (x) = 1 + 1 − e−θx 1 + , x > 0. 2 1+θ

Corollary 2.283. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.142. The random variable X has pd f (1.1.896) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = ξ (x) q1 (x) − q2 (x)

αθ2 −θx 1+θ (1 + x) e

θx 1 − e−θx 1 + 1+θ 
α θx 1 − 1 − e−θx 1 + 1+θ




α−1

, x > 0.

Corollary 2.284. The general solution of the differential equation in Corollary 2.283 is    α −1 θx −θx ξ (x) = 1 − 1 − e 1+ × 1+θ " Z #   α−1 αθ2 θx −1 −θx −θx − (1 + x) e 1−e 1+ (q1 (x)) q2 (x) dx + D , 1+θ 1+θ where D is a constant.

Characterizations of Distributions

353

Proposition 2.143. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =
−1 θ + θα−1 xα−1 and q2 (x) = q1 (x) e−θx , x > 0. The random variable X has pd f (1.1.900) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−θx , x > 0. 2 Corollary 2.285. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.143. The random variable X has pd f (1.1.900) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = θ, x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.286. The general solution of the differential equation in Corollary 2.285 is  Z  −1 θx −θx ξ (x) = e − θe (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.144. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  )−1  ( 2 2 (θ + α) α + θ 2x and q2 (x) = q1 (x) e−θx , x > 0. The random variable X has +θ (θ − 1) (1 + αx) pd f (1.1.906) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−θx , x > 0. 2 Corollary 2.287. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.144. The random variable X has pd f (1.1.906) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = θ, x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.288. The general solution of the differential equation in Corollary 2.287 is  Z  −1 θx −θx ξ (x) = e − θe (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.145. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =
−1 x−c θ + x2 and q2 (x) = q1 (x) e−θx , x > 0. The random variable X has pd f (1.1.910) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−θx , x > 0. 2

G. G. Hamedani

354

Corollary 2.289. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.145. The random variable X has pd f (1.1.910)if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = θ, x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.290. The general solution of the differential equation in Corollary 2.289 is  Z  −1 θx −θx ξ (x) = e − θe (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.146. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = δ 2 x−2δ−β+2 and q2 (x) = q1 (x) e− ω x , x > 0. The random variable X has pd f (1.1.914) if and only if the function ξ defined in Theorem 2.1 is of the form 1 δ 2 ξ (x) = e− ω x , x > 0. 2 Corollary 2.291. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.146. The random variable X has pd f (1.1.914) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation 2δ ξ0 (x) q1 (x) = x, x > 0. ξ (x) q1 (x) − q2 (x) ω Corollary 2.292. The general solution of the differential equation in Corollary 2.291 is  Z  δ 2 2δ − δ x2 ξ (x) = e ω x − xe ω (q1 (x))−1 q2 (x) dx + D , ω where D is a constant. Proposition 2.147. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = β −β x−θ and q2 (x) = q1 (x) e−λα x , x > 0. The random variable X has pd f (1.1.916) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

 1 β −β 1 + e−λα x , x > 0. 2

Corollary 2.293. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.147. The random variable X has pd f (1.1.916) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) λβαβ x−β−1 e−λα = ξ (x) q1 (x) − q2 (x) 1 − e−λαβ x−β

β x−β

, x > 0.

Characterizations of Distributions

355

Corollary 2.294. The general solution of the differential equation in Corollary 2.293 is     Z β −β −1 β −β ξ (x) = 1 − e−λα x − λβαβ x−β−1 e−λα x (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

Proposition 2.148. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = 1 θx x 2 −c and q2 (x) = q1 (x) e− 2 , x > 0. The random variable X has pd f (1.1.918) if and only if the function ξ defined in Theorem 2.1 is of the form 1 θx ξ (x) = e− 2 , x > 0. 2 Corollary 2.295. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.148. The random variable X has pd f (1.1.918) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) θ = , x > 0. ξ (x) q1 (x) − q2 (x) 2 Corollary 2.296. The general solution of the differential equation in Corollary 2.295 is  Z  θx θ − θx −1 ξ (x) = e 2 − e 2 (q1 (x)) q2 (x) dx + D , 2 where D is a constant. Proposition 2.149. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = n 
α o1−β 
 1 − 1 + θx e−x/θ and q2 (x) = q1 (x) 1 + θx e−x/θ , x > 0. The random variable X has pd f (1.1.920) if and only if the function ξ defined in Theorem 2.1 is of the form α h x  −x/θ i 1+ e , x > 0. ξ (x) = α+1 θ

Corollary 2.297. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.149. The random variable X has pd f (1.1.920) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) αθ−2 x = , x > 0. ξ (x) q1 (x) − q2 (x) 1 + θx Corollary 2.298. The general solution of the differential equation in Corollary 2.297 is  Z  h x  −x/θ i−1 −1 −2 −x/θ ξ (x) = 1 + e − αθ xe (q1 (x)) q2 (x) dx + D , θ where D is a constant.

G. G. Hamedani

356

Proposition 2.150. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =
−1 x + x5 and q2 (x) = q1 (x) e−αx, x > 0. The random variable X has pd f (1.1.922) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−αx , x > 0. 2 Corollary 2.299. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.150. The random variable X has pd f (1.1.922) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = α, x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.300. The general solution of the differential equation in Corollary 2.299 is  Z  −1 αx −αx ξ (x) = e − αe (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.151. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = −1 [(1−λ−β)+2(λ+2β)e−mx −3βe−2mx ] and q2 (x) = q1 (x) e−mx, x > 0. The random variable −mx −2mx −3mx a−1 [1+(λ+β−1)e

−(λ+2β)e

+βe

]

X has pd f (1.1.926) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−mx , x > 0. 2

Corollary 2.301. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.151. The random variable X has pd f (1.1.926) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = m, x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.302. The general solution of the differential equation in Corollary 2.301 is  Z  ξ (x) = emx − me−mx (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.152. Let X : Ω →(0, ∞) be a continuous random variable and let q1 (x) =  −λ/x2 −λ/x2 2 β (α + 1) + (α − 1) − 2βαe α−e and q2 (x) = q1 (x) e−λ/x , x > 0. The random

variable X has pd f (1.1.928) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

 1 2 1 + e−λ/x , x > 0. 2

Characterizations of Distributions

357

Corollary 2.303. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.152. The random variable X has pd f (1.1.928) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation 2

ξ0 (x) q1 (x) 2λe−λ/x
, x > 0. = 3 ξ (x) q1 (x) − q2 (x) x 1 − e−λ/x2

Corollary 2.304. The general solution of the differential equation in Corollary 2.303 is " Z # 2  −1 2λe−λ/x −1 −λ/x2 ξ (x) = 1 − e − (q1 (x)) q2 (x) dx + D , x3 where D is a constant. Proposition 2.153. Let X : Ω → R be a continuous random variable and let q1 (x) =  1 + (logβ) βG(x;η) and q2 (x) = q1 (x) G (x; η) , x ∈ R. The random variable X has pd f (1.1.930) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 (1 + G (x; η)) , x ∈ R. 2

Corollary 2.305. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.153. The random variable X has pd f (1.1.930) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) g (x; η) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; η) Corollary 2.306. The general solution of the differential equation in Corollary 2.305 is  Z  −1 −1 ξ (x) = (1 − G (x; η)) − g (x; η) (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.154. Let X : Ω → R be a continuous random variable and let q1 (x) =   −b 1−a −b 1 − e1−(1−G(x;η)) and q2 (x) = q1 (x) e1−(1−G(x;η)) , x ∈ R. The random variable X has pd f (1.1.938) if and only if the function ξ defined in Theorem 2.1 is of the form −b 1 ξ (x) = e1−(1−G(x;η)) , x ∈ R. 2

Corollary 2.307. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.154. The random variable X has pd f (1.1.938) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = bg (x; η) (1 − G (x; η))−b−1 , x ∈ R. ξ (x) q1 (x) − q2 (x)

G. G. Hamedani

358

Corollary 2.308. The general solution of the differential equation in Corollary 2.307 is −b

ξ (x) = e−1+(1−G(x;η)) ×  Z  −b − bg (x; η) (1 − G (x; η))−b−1 e1−(1−G(x;η)) (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.155. Let X : Ω → R be a continuous random variable and let q1 (x) = i−1  −1 h  λ G(x) and q2 (x) = q1 (x) 1 − λG (x) , x ∈ R. The random variable X has v 1 −λG(x)

pd f (1.1.944) if and only if the function ξ defined in Theorem 2.1 is of the form  −1  −1  1  ξ (x) = 1 −λ + 1 − λG (x) , x ∈ R. 2

Corollary 2.309. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.155. The random variable X has pd f (1.1.944) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation  −2 λg (x) 1 − λG (x) 1 (x) = −1  −1 , x ∈ R. ξ (x) q1 (x) − q2 (x) + 1 − λG (x) 1 −λ ξ0 (x) q

Corollary 2.310. The general solution of the differential equation in Corollary 2.309 is  −1  −1 −1 ξ (x) = 1 −λ + 1 − λG (x) ×  Z   −2 −1 − λg (x) 1 − λG (x) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.156. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = ) (  exp e

(

βxα − σ2 x

2

  2θx+ αβxα−1 + 2σ3 e x

 ) βxα − σ2 x

and q2 (x) = q1 (x) e−θx , x > 0. The random variable X has

pd f (1.1.946) if and only if the function ξ defined in Theorem 2.1 is of the form 1 2 ξ (x) = e−θx , x > 0. 2 Corollary 2.311. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.156. The random variable X has pd f (1.1.946) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) θx = , x > 0. ξ (x) q1 (x) − q2 (x) 2

Characterizations of Distributions

359

Corollary 2.312. The general solution of the differential equation in Corollary 2.311 is  Z  2 θx −θx2 e (q1 (x))−1 q2 (x) dx + D , ξ (x) = eθx − 2 where D is a constant. Proposition 2.157. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = (1 − α cos (βx))−1 and q2 (x) = q1 (x) e−λx , x > 0. The random variable X has pd f (1.1.948) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−λx , x > 0. 2 Corollary 2.313. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.157. The random variable X has pd f (1.1.948) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = λ, x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.314. The general solution of the differential equation in Corollary 2.313 is  Z  −1 λx −λx ξ (x) = e − λe (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.158. Let X : Ω → R be a continuous random variable and let q1 (x) = ζ+1 {1+e−(υx+sign(x)(η/υ)|x|)} −υx , x ∈ R. The random variable X has pd f υ−1 −sign(x)(η/υ)|x|υ and q2 (x) = q1 (x) e (υ+η|x| )e (1.1.950) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−υx , x ∈ R. 2 Corollary 2.315. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.158. The random variable X has pd f (1.1.950) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = υ, x ∈ R. ξ (x) q1 (x) − q2 (x) Corollary 2.316. The general solution of the differential equation in Corollary 2.315 is  Z  −1 υx −υx ξ (x) = e − υe (q1 (x)) q2 (x) dx + D , where D is a constant.

G. G. Hamedani

360

Proposition 2.159. Let X : Ω → R be a continuous random variable and let q1 (x) ≡ 1 and q2 (x) = (1 − λG (x)) eλG(x) , x ∈ R. The random variable X has pd f (1.1.952) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

o 1n (1 − λG (x)) eλG(x) + (1 − λ) eλ , x ∈ R. 2

Corollary 2.317. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.159. The random variable X has pd f (1.1.952) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation λ2 g (x) G (x) eλG(x) ξ0 (x) q1 (x) = , x ∈ R. ξ (x) q1 (x) − q2 (x) (1 − λG (x)) eλG(x) − (1 − λ) eλ Corollary 2.318. The general solution of the differential equation in Corollary 2.317 is n o−1 ξ (x) = (1 − λG (x))eλG(x) − (1 − λ) eλ ×  Z  −1 2 λG(x) − λ g (x) G (x) e (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.160. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = h i− 1

2 γ γ 1 − 1+γ 1 − θ + θe−λx e−λx and q2 (x) = q1 (x) 1 − θ + 2θe−λx , x > 0. The random variable X has pd f (1.1.954) if and only if the function ξ defined in Theorem 2.1 is of the form   2 1  ξ (x) = 1 − θ + 2θe−λx + (1 − θ)2 , x > 0. 2

Corollary 2.319. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.160. The random variable X has pd f (1.1.954) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation
2 4λ 1 − θ + 2θe−λx e−λx ξ0 (x) q1 (x) = , x > 0.
2 ξ (x) q1 (x) − q2 (x) 1 − θ + 2θe−λx − (1 − θ)2

Corollary 2.320. The general solution of the differential equation in Corollary 2.319 is  −1 2 2 −λx 1 − θ + 2θe − (1 − θ) × ξ (x) =  Z   2 − 4λ 1 − θ + 2θe−λx e−λx (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

Characterizations of Distributions

361

Proposition 2.161. Let X : Ω → R be a continuous random variable and let q1 (x) = k+1 [1 + G (x; η)c ] and q2 (x) = q1 (x) G (x; η)c , x ∈ R. The random variable X has pd f (1.1.958) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 {1 + G (x; η)c }, x ∈ R. 2

Corollary 2.321. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.161. The random variable X has pd f (1.1.958) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation cg (x; η) G (x; η)c−1 ξ0 (x) q1 (x) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; η)c Corollary 2.322. The general solution of the differential equation in Corollary 2.321 is  Z  c −1 c−1 −1 ξ (x) = {1 − G (x; η) } − cg (x; η) G (x; η) (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.162. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  (    −β ) −1 −α    fT QY 1+ xγ  h  α i−1 x α(1−β) (  ) , x > 0. The random x and q (x) = q (x) 1 +  2 1  −α −β γ    fY QY 1+ xγ  variable X has pd f (1.1.966) if and only if the function ξ defined in Theorem 2.1 is of the form   α −1 β x 1+ , x > 0. ξ (x) = β+1 γ Corollary 2.323. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.162. The random variable X has pd f (1.1.966) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation 0

αβ γ

 α−1 x

γ ξ (x) q1 (x) =h  α i, x > 0. ξ (x) q1 (x) − q2 (x) 1 + xγ

Corollary 2.324. The general solution of the differential equation in Corollary 2.323 is #   α −1 " Z  α−1 αβ x x −1 ξ (x) = 1 + − (q1 (x)) q2 (x) dx + D , γ γ γ where D is a constant.

G. G. Hamedani

362

Proposition 2.163. Let X : Ω → R be a continuous random variable and let q1 (x) = exp{λ[1−G(x;η)][1+γG(x;η)] / G(x;η)} and q2 (x) = q1 (x) G (x; η)−1 , x ∈ R. The random variable [1+γG(x;η)2 ] X has pd f (1.1.968) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

i 1h 1 + G (x; η)−1 , x ∈ R. 2

Corollary 2.325. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.163. The random variable X has pd f (1.1.968) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation g (x; η) G (x; η)−2 ξ0 (x) q1 (x) i , x ∈ R. = h ξ (x) q1 (x) − q2 (x) 1 − G (x; η)−1

Corollary 2.326. The general solution of the differential equation in Corollary 2.325 is  h i−1  Z −1 −2 −1 ξ (x) = 1 − G (x; η) − g (x; η) G (x; η) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.164. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) = n h io fY QY 1−(1−xα )β n h io fT QY 1−(1−xα )β

and q2 (x) = q1 (x) (1 − xα ) , 0 < x < 1. The random variable X has pd f

(1.1.982) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

β (1 − xα ), 0 < x < 1. β+1

Corollary 2.327. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.164. The random variable X has pd f (1.1.982) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) αβxα−1 = , 0 < x < 1. ξ (x) q1 (x) − q2 (x) 1 − xα

Corollary 2.328. The general solution of the differential equation in Corollary 2.327 is  Z  ξ (x) = (1 − xα )−1 − αβxα−1 (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.165. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = (1+xα )−1 n   oa−1 βxα − log 1+ β+1 +βxα

α

and q2 (x) = q1 (x) e−βx , x > 0. The random variable X has pd f

(1.1.984) if and only if the function ξ defined in Theorem 2.1 is of the form 1 α ξ (x) = e−βx , x > 0. 2

Characterizations of Distributions

363

Corollary 2.329. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.165. The random variable X has pd f (1.1.984) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = αβxα−1 , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.330. The general solution of the differential equation in Corollary 2.329 is  Z  −1 βxα α−1 −βxα ξ (x) = e − αβx e (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.166. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = 2 x (2Φ (x) − 1)−1 and q2 (x) = q1 (x) e−x /2 , x > 0. The random variable X has pd f (1.1.988) if and only if the function ξ defined in Theorem 2.1 is of the form 1 2 ξ (x) = e−x /2 , x > 0. 2 Corollary 2.331. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.166. The random variable X has pd f (1.1.988) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = x, x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.332. The general solution of the differential equation in Corollary 2.331 is  Z  −1 x2 /2 −x2 /2 ξ (x) = e − xe (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.167. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = h i1−α 1 − (1 − βλx)1/β and q2 (x) = q1 (x) (1 − βλx)1/β , x > 0. The random variable X has pd f (1.1.992) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 (1 − βλx)1/β , x > 0. 2

Corollary 2.333. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.167. The random variable X has pd f (1.1.992) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) λ = , x > 0. ξ (x) q1 (x) − q2 (x) 1 − βλx

G. G. Hamedani

364

Corollary 2.334. The general solution of the differential equation in Corollary 2.333 is  Z  ξ (x) = (1 − βλx)−1 − λ (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.168. Let X : Ω → R be a continuous random variable and let q1 (x) = h i−1 1 + (logβ) βG(x;ζ) and q2 (x) = q1 (x) G (x; ζ) , x > 0. The random variable X has pd f (1.1.998) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 (1 + G (x; ζ)), x ∈ R. 2

Corollary 2.335. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.168. The random variable X has pd f (1.1.998) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) g (x; ζ) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; ζ) Corollary 2.336. The general solution of the differential equation in Corollary 2.335 is  Z  −1 −1 ξ (x) = (1 − G (x; ζ)) − g (x; ζ) (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.169. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = n



a−1 

b−1 o−1 C0 θ 1 − e−βx C θ 1 − e−βx C (θ) −C θ 1 − e−βx and q2 (x) =

q1 (x) e−βx , x > 0. The random variable X has pd f (1.1.1002) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−βx , x > 0. 2 Corollary 2.337. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.169. The random variable X has pd f (1.1.1006) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = β, x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.338. The general solution of the differential equation in Corollary 2.337 is  Z  −1 βx −βx ξ (x) = e − βe (q1 (x)) q2 (x) dx + D , where D is a constant.

Characterizations of Distributions

365

Proposition 2.170. Let X : Ω → R be a continuous random variable and let q1 (x) =  1−b h i−2 2 δ2 G (x;η) 1 1 − and q (x) = q (x) 1 − δG (x; η) , x ∈ R. The random vari2 1 2 G(x;η) [1−δG(x;η)] able X has pd f (1.1.1016) if and only if the function ξ defined in Theorem 2.1 is of the form  i2  h 1 , x ∈ R. ξ (x) = 1 + 1 − δG (x; η) 2

Corollary 2.339. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.170. The random variable X has pd f (1.1.1016) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation h i−2 2δg (x; η) 1 − δG (x; η) ξ (x) q1 (x) =− , x ∈ R. h i−2 ξ (x) q1 (x) − q2 (x) 1 − δG(x; η) 0

Corollary 2.340. The general solution of the differential equation in Corollary 2.339 is  h i−2 −1 ξ (x) = 1 − 1 − δG (x; η) ×  Z h i−2 −1 (q1 (x)) q2 (x) dx + D , 2δg (x; η) 1 − δG(x; η)

where D is a constant.

Proposition 2.171. Let X : Ω → R be a continuous random variable and let q1 (x) = h i−1 1 − θ + 2θ (1 − G (x; η))λ and q2 (x) = q1 (x) [1 − G (x; η)] , x ∈ R. The random variable X has pd f (1.1.1034) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

λ [1 − G (x; η)], x ∈ R. λ+1

Corollary 2.341. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.171. The random variable X has pd f (1.1.1034) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) λg (x; η) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; η) Corollary 2.342. The general solution of the differential equation in Corollary 2.341 is  Z  −1 −1 ξ (x) = [1 − G (x; η)] − λg (x; η) (q1 (x)) q2 (x) dx + D , where D is a constant.

G. G. Hamedani

366

Proposition 2.172. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = o n β(λx)β−1 and q2 (x) = q1 (x) e−λx , x > 0. The random variable X has pd f (1.1.1036) 1 + Γ(β) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−λx , x > 0. 2 Corollary 2.343. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.172. The random variable X has pd f (1.1.1036) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = λ, x > 0. ξ (x) q1 (x) − q2 (x)

Corollary 2.344. The general solution of the differential equation in Corollary 2.343 is  Z  −1 λx −λx ξ (x) = e − λe (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.173. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =    γ   2   β β  β  λ 1−ex δλ 1−ex (1−δ)λ 1−ex 1−e  +e e     #)"   #γ−1 ( " β β γ+(δ−γ) 1−e

λ 1−ex

1−e

λ 1−ex

  β λ 1−ex

and q2 (x) = q1 (x) e

, x > 0. The random

variable X has pd f (1.1.1042) if and only if the function ξ defined in Theorem 2.1 is of the form 1 λ 1−exβ ξ (x) = e , x > 0. 2 Corollary 2.345. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.173. The random variable X has pd f (1.1.1042) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation 



ξ0 (x) q1 (x) β = λβxβ−1 ex , x > 0. ξ (x) q1 (x) − q2 (x)

Corollary 2.346. The general solution of the differential equation in Corollary 2.345 is    Z   β xβ −λ 1−ex −1 β−1 xβ λ 1−e ξ (x) = e − λβx e e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.174. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = β x−c and q2 (x) = q1 (x) e−θx , x > 0. The random variable X has pd f (1.1.1054) if and 1+xβ only if the function ξ defined in Theorem 2.1 is of the form 1 β ξ (x) = e−θx , x > 0. 2

Characterizations of Distributions

367

Corollary 2.347. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.174. The random variable X has pd f (1.1.1054) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = θβxβ−1 , x > 0. ξ (x) q1 (x) − q2 (x)

Corollary 2.348. The general solution of the differential equation in Corollary 2.347 is  Z  −1 θxβ β−1 −θxβ − θβx e (q1 (x)) q2 (x) dx + D , ξ (x) = e

where D is a constant.

Proposition 2.175. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = h

2 i β γ β 1+( λx ) −α e−x h i β γ−1 1+( λx )

β

and q2 (x) = q1 (x) e−x , x > 0.

The random variable X has pd f

(1.1.1056) if and only if the function ξ defined in Theorem 2.1 is of the form 1 β ξ (x) = e−x , x > 0. 2 Corollary 2.349. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.175. The random variable X has pd f (1.1.1056) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = βxβ−1 , x > 0. ξ (x) q1 (x) − q2 (x)

Corollary 2.350. The general solution of the differential equation in Corollary 2.349 is  Z  −1 xβ β−1 −xβ ξ (x) = e − βx e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.176. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = h  i exp − 12 logβ x +2 2α  1/2  3/2  β − log x + logβ x

log x

and q2 (x) = q1 (x) e 2α2 β , x > 0.

The random variable X has pd f

(1.1.1064) if and only if the function ξ defined in Theorem 2.1 is of the form   log x 1 2β 2α ξ (x) = 1+e , x > 0. 2

Corollary 2.351. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.176. The random variable X has pd f (1.1.1064) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = ξ (x) q1 (x) − q2 (x)

log x

x−1 e 2α2 β   , x > 0. log x 2β 2 2α 2α β 1 − e

G. G. Hamedani

368

Corollary 2.352. The general solution of the differential equation in Corollary 2.351 is   log x −1  Z −1 2α 2β log x x e − (q1 (x))−1 q2 (x) dx + D , ξ (x) = 1 − e 2α2 β 2α2 β

where D is a constant.

Proposition 2.177. Let X : Ω → R be a continuous random variable and let q1 (x) =  2 G(x;θ) G (x; θ)2 exp G(x;θ) and q2 (x) = q1 (x) G (x; θ), x ∈ R. The random variable X has pd f (1.1.1066) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 {1 + G (x; θ)} , x ∈ R. 2

Corollary 2.353. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.177. The random variable X has pd f (1.1.1066) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) g (x; θ) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; θ) Corollary 2.354. The general solution of the differential equation in Corollary 2.353 is  Z  −1 −1 ξ (x) = (1 − G (x; θ)) − g (x; θ)(q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 
2.178. Let X : Ω → (0,−21) be a continuous random variable and let q1 (x) = exp θ 1−x and q2 (x) = q1 (x) x , 0 < x < 1. The random variable X has pd f x (1.1.1068) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 1 + x−2 , 0 < x < 1. 2

Corollary 2.355. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.178. The random variable X has pd f (1.1.1068) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation 2x−3 ξ0 (x) q1 (x) =− , 0 < x < 1. ξ (x) q1 (x) − q2 (x) 1 − x−2 Corollary 2.356. The general solution of the differential equation in Corollary 2.355 is Z 
−1 −2 −1 −3 ξ (x) = 1 − x 2x (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Characterizations of Distributions

369

Proposition 2.179. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = (

)1−β 2   α θxα (θxα +2) −2θx e 1− 1− 1+ θ3 +2   θxα (θxα +2) α (θ+x2α ) 1− e−θx 3

α

and q2 (x) = q1 (x) e−θx , x > 0. The random variable X

θ +2

has pd f (1.1.1086) if and only if the function ξ defined in Theorem 2.1 is of the form 1 α ξ (x) = e−θx , x > 0. 2 Corollary 2.357. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.179. The random variable X has pd f (1.1.1086) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = αθxα−1 , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.358. The general solution of the differential equation in Corollary 2.357 is  Z −1 α−1 −θxα θxα ξ (x) = e αθx e (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.180. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = − m x2(m−1)−a and q2 (x) = q1 (x) e wx2 , x > 0. The random variable X has pd f (1.1.1088) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

o 1n − m 1 + e wx2 , x > 0. 2

Corollary 2.359. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.180. The random variable X has pd f (1.1.1088) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation −

m

2m −3 x e wx2 ξ0 (x) q1 (x) = w , x > 0. − m ξ (x) q1 (x) − q2 (x) 1 − e wx2

Corollary 2.360. The general solution of the differential equation in Corollary 2.359 is  n o Z 2m − m2 −1 − m2 −1 −3 ξ (x) = 1 − e wx x e wx (q1 (x)) q2 (x) dx + D , w where D is a constant. Proposition 2.181. Let X : Ω → R be a continuous random variable and let q1 (x) =  1−α − logG (x) and q2 (x) = q1 (x) G (x) , x ∈ R. The random variable X has pd f (1.1.1094) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

β G (x) , x ∈ R. β+1

G. G. Hamedani

370

Corollary 2.361. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.181. The random variable X has pd f (1.1.1094) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation βg (x) ξ0 (x) q1 (x) = , x ∈ R. ξ (x) q1 (x) − q2 (x) G (x) Corollary 2.362. The general solution of the differential equation in Corollary 2.361 is  Z   −1 −1 ξ (x) = G(x) − βg (x) (q1 (x)) q2 (x) dx + D , where D is a constant.

Proposition 2.182. Let X : Ω → R be a continuous random variable and let q1 (x) = exp [ace−x] exp [−ce−x ] and q2 (x) = q1 (x) (1 − exp [−ce−x ]) , x ∈ R. The random variable X has pd f (1.1.1098) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

  b  1 − exp −ce−x , x ∈ R. b+1

Corollary 2.363. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.182. The random variable X has pd f (1.1.1098) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) bce−x exp [−ce−x ] = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − exp [−ce−x ] Corollary 2.364. The general solution of the differential equation in Corollary 2.363 is  Z       −1 −x −1 −x −x ξ (x) = 1 − exp −ce − bce exp −ce (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.183. Let X : Ω → R be a continuous random variable and let q1 (x) = h i−1
2 −1 1 − αx − βx3 + 1 and q2 (x) = q1 (x) (1 − e−x ) , x ∈ R. The random variable X has pd f (1.1.1100) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =


−1 o 1n 1 + 1 − e−x , x ∈ R. 2

Corollary 2.365. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.183. The random variable X has pd f (1.1.1100) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation −2

ξ0 (x) q1 (x) e−x (1 − e−x ) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − (1 − e−x )−1

Characterizations of Distributions

371

Corollary 2.366. The general solution of the differential equation in Corollary 2.365 is  Z  n
−1 o−1
−2 ξ (x) = 1 − 1 − e−x − e−x 1 − e−x (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

Proposition 2.184. random variable and let q1 (x) =  Let X : Ω →R be a continuous  c c 2 c 3 aG(x;φ) G(x;φ) (1−G(x;φ) ) exp + b2 1−G(x;φ) and q2 (x) = q1 (x) G (x; φ)c , x ∈ R. The c a+(b−a)G(x;φ)c 1−G(x;φ)c random variable X has pd f (1.1.1106) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 {1 + G (x; φ)c } , x ∈ R. 2

Corollary 2.367. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.184. The random variable X has pd f (1.1.1106) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation cg (x; φ) G (x; φ)c−1 ξ0 (x) q1 (x) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; φ)c Corollary 2.368. The general solution of the differential equation in Corollary 2.367 is  Z  c −1 c−1 −1 − cg (x; φ)G (x; φ) (q1 (x)) q2 (x) dx + D , ξ (x) = {1 − G (x; φ) } where D is a constant. Proposition 2.185. Let X :Ω → R be acontinuous random variable and let q1 (x) = (1−G(x;κ))

 β G(x;κ) β+1 − 1−G(x;κ)



e





1−exp−αe



expαe

β G(x;κ) 1−G(x;κ)



β G(x;κ) 1−G(x;κ)

θ−1

−1−

and q2 (x) = q1 (x) G (x; κ)β , x ∈ R.

The

−1

random variable X has pd f (1.1.1108) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

o 1n 1 + G (x; κ)β , x ∈ R. 2

Corollary 2.369. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.185. The random variable X has pd f (1.1.1108) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) βg (x; κ)G (x; κ)β−1 = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; κ)β

372

G. G. Hamedani

Corollary 2.370. The general solution of the differential equation in Corollary 2.369 is  n o−1  Z ξ (x) = 1 − G (x; κ)β − βg (x; κ) G (x; κ)β−1 (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

Proposition Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = h   2.186.β i β cosh α 1 − e−λx and q2 (x) = q1 (x) e−λx , x > 0. The random variable X has pd f

(1.1.1112) if and only if the function ξ defined in Theorem 2.1 is of the form β 1 ξ (x) = e−λx , x > 0. 2

Corollary 2.371. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.186. The random variable X has pd f (1.1.1112) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = βλxβ−1 , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.372. The general solution of the differential equation in Corollary 2.371 is  Z  −1 λxβ β−1 −λxβ ξ (x) = e − βλx e (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.187. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =   β k+1 β 1 − pe−(xα) and q2 (x) = q1 (x) e−k(xα) , x > 0. The random variable X has pd f (1.1.1118) if and only if the function ξ defined in Theorem 2.1 is of the form β 1 ξ (x) = e−k(xα) , x > 0. 2

Corollary 2.373. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.187. The random variable X has pd f (1.1.1118) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = βαβ kxβ−1 , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.374. The general solution of the differential equation in Corollary 2.373 is  Z  β β ξ (x) = ek(xα) − βαβ kxβ−1 e−k(xα) (q1 (x))−1 q2 (x) dx + D , where D is a constant.

Characterizations of Distributions

373

Proposition 2.188. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) = 2α 1 log(x) e 2 (− log(x)) and q2 (x) = q1 (x) log (x) , 0 < x < 1. The random variable X has pd f (1.1.1120) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 log(x) , 0 < x < 1. 2

Corollary 2.375. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.188. The random variable X has pd f (1.1.1120) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) 1 =− , 0 < x < 1. ξ (x) q1 (x) − q2 (x) x log(x) Corollary 2.376. The general solution of the differential equation in Corollary 2.375 is Z  −1 −1 −1 ξ (x) = (log(x)) x (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.189. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) ≡ 1 !−1 −1/λ −θ+θ(1−λxθ ) −1 , x > 0. The random variable X has pd f and q2 (x) = 1 + e eθ −1 (1.1.1126) if and only if the function ξ defined in Theorem 2.1 is of the form

ξ (x) =



−1/λ

−θ+θ(1−λxθ )

β  e 1+ β+1

eθ − 1

−1

−1

, x > 0.

Corollary 2.377. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.189. The random variable X has pd f (1.1.1126) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation

ξ0 (x) q1 (x) = ξ (x) q1 (x) − q2 (x)

θ2



−θ+θ(1−λxθ )

e



−1/λ

xθ−1

θ eθ + e−θ+θ(1−λx )


− 1 −1 1 − λxθ λ

−1/λ

−2





, x > 0.

Corollary 2.378. The general solution of the differential equation in Corollary 2.377 is  −1 −1/λ −θ+θ( 1−λxθ ) θ ξ (x) = e + e −2 ×  Z    − 1 −1  −1/λ λ −1 −θ+θ(1−λxθ ) 2 θ−1 θ − θ e x 1 − λx (q1 (x)) q2 (x) dx + D ,

where D is a constant.

G. G. Hamedani

374

Proposition 2.190. Let X : Ω → R be a continuous random variable and let q1 (x) =  1 − θG (x; η) and q2 (x) = q1 (x) eθG(x;η) , x ∈ R. The random variable X has pd f (1.1.1138) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

o 1 n θG(x;η) e + 1 , x ∈ R. 2

Corollary 2.379. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.190. The random variable X has pd f (1.1.1138) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation 1 g (x; η) eθG(x;η) ξ0 (x) q1 (x) , x ∈ R. = θ ξ (x) q1 (x) − q2 (x) eθG(x;η) − 1

Corollary 2.380. The general solution of the differential equation in Corollary 2.379 is  n o−1  Z 1 −1 θG(x;η) θG(x;η) ξ (x) = e −1 − g (x; η) e (q1 (x)) q2 (x) dx + D , θ where D is a constant.

Proposition 2.191. Let X : Ω → R be a continuous random variable and let q1 (x) = ( f T [aG (x)])−1 and q2 (x) = q1 (x) G (x) , x ∈ R. The random variable X has pd f (1.1.1144) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 {1 + G (x)}, x ∈ R. 2

Corollary 2.381. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.191. The random variable X has pd f (1.1.1144) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) g (x) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x) Corollary 2.382. The general solution of the differential equation in Corollary 2.381 is  Z  −1 −1 ξ (x) = {1 − G (x)} − g (x) (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition random variable and let q1 (x) = h  −δ 2.192. i Let X : Ω → (0, ∞)  be−δ a continuous  x x exp b e − 1 and q2 (x) = q1 (x) e − 1 , x > 0. The random variable X has pd f

(1.1.1156) if and only if the function ξ defined in Theorem 2.1 is of the form  a  x−δ ξ (x) = e − 1 , x > 0. a+1

Characterizations of Distributions

375

Corollary 2.383. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.192. The random variable X has pd f (1.1.1156) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation −δ

δax−(δ+1) ex ξ0 (x) q1 (x) , x > 0. = −δ ξ (x) q1 (x) − q2 (x) ex − 1 Corollary 2.384. The general solution of the differential equation in Corollary 2.383 is   −δ −1  Z −1 x −(δ+1) x−δ ξ (x) = e − 1 − δax e (q1 (x)) q2 (x) dx + D ,

where D is a constant.

Proposition 2.193. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  α   β  −α  1+ x −1 β e , x > 0. The random variable X has pd f  α(θ+1) and q2 (x) = q1 (x) 1 − 1 + x 1+ xβ

(1.1.1196) if and only if the function ξ defined in Theorem 2.1 is of the form "  −α # θ β ξ (x) = 1− 1+ , x > 0. θ+1 x

Corollary 2.385. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.193. The random variable X has pd f (1.1.1196) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation  −α−1 β αθ 1 + x ξ (x) q1 (x) =  −α , x > 0. ξ (x) q1 (x) − q2 (x) 1 − 1 + βx 0

Corollary 2.386. The general solution of the differential equation in Corollary 2.385 is " #  #−1 " Z    β −α β −α−1 −1 ξ (x) = 1 − 1 + − αθ 1 + (q1 (x)) q2 (x) dx + D , x x where D is a constant. Proposition 2.194. Let X : Ω → (lnk, ∞) be a continuous random variable and let q1 (x) =  −1 3 − λ2 − λ1 + 2 (2λ2 + λ1 − 3) ka e−ax + 3 (1 − λ2 ) k2a e−2ax and q2 (x) = q1 (x) e−ax, x > lnk. The random variable X has pd f (1.1.1204) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−ax , x > ln k. 2 Corollary 2.387. Let X : Ω → (ln k, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.194. The random variable X has pd f (1.1.1204) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = a, x > ln k. ξ (x) q1 (x) − q2 (x)

G. G. Hamedani

376

Corollary 2.388. The general solution of the differential equation in Corollary 2.387 is  Z  ξ (x) = eax − ae−ax (q1 (x))−1 q2 (x) dx + D , where D is a constant. Proposition 2.195. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = −1    θ p x 1 , θ + 10 , 1 and q2 (x) = q1 (x) x−1 , x > 0. The random variable X has pd f F β (1.1.1208) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

p −1 x , x > 0. p+1

Corollary 2.389. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.195. The random variable X has pd f (1.1.1208) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = px−1 , x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.390. The general solution of the differential equation in Corollary 2.389 is  Z  −1 −1 ξ (x) = x − p (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.196.Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = 1 log x, 0 0.

Corollary 2.396. The general solution of the differential equation in Corollary 2.395 is  Z  −1 p −(p+1) ξ (x) = x − px (q1 (x)) q2 (x) dx + D , where D is a constant.

G. G. Hamedani

378

Proposition 2.199. Let X : Ω → R be a continuous random variable and let q1 (x) = i h  h ik −1 G(x;η) −λ G(x;η) G(x;η) s , x ∈ R. The random variable X has a and q (x) = q (x) e ∑k=0 k G(x;η) 2 1 pd f (1.1.1218) if and only if the function ξ defined in Theorem 2.1 is of the form 1 −λ ξ (x) = e 2

h

G(x;η) G(x;η)

i

x ∈ R.

,

Corollary 2.397. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.199. The random variable X has pd f (1.1.1218) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) g (x; η) = λ 2 , ξ (x) q1 (x) − q2 (x) G (x; η)

x ∈ R.

Corollary 2.398. The general solution of the differential equation in Corollary 2.397 is # i" Z i h h G(x;η) G(x;η) g (x; η) −λ G(x;η) λ G(x;η) −1 − λ (q1 (x)) q2 (x) dx + D , ξ (x) = e 2 e G (x; η) where D is a constant.

Proposition 2.200. Let X : Ω → R be a continuous random variable and let q1 (x) = h  i−1 x−µ σ δσ ΦB x−µ − , ; −δ and q2 (x) = q1 (x) e−( σ ) , x ∈ R. The random variable X has σ τ τ pd f (1.1.1230) if and only if the function ξ defined in Theorem 2.1 is of the form x−µ 1 ξ (x) = e−( σ ) , 2

x ∈ R.

Corollary 2.399. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.200. The random variable X has pd f (1.1.1230) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) 1 = , ξ (x) q1 (x) − q2 (x) σ

x ∈ R.

Corollary 2.400. The general solution of the differential equation in Corollary 2.399 is  Z  x−µ 1 −( x−µ ) −1 ( ) ξ (x) = e σ − e σ (q1 (x)) q2 (x) dx + D , σ where D is a constant. Proposition 2.201. Let X : Ω → R be a continuous random variable and let q1 (x) = h i−1 2λ 1−λ+ and q2 (x) = q1 (x) G (x) , x ∈ R. The random variable X has pd f 2 (1+G(x))

(1.1.1234) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

1 (1 + G (x)) , 2

x ∈ R.

Characterizations of Distributions

379

Corollary 2.401. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.201. The random variable X has pd f (1.1.1234) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation g (x) ξ0 (x) q1 (x) = , ξ (x) q1 (x) − q2 (x) 1 − G (x)

x ∈ R.

Corollary 2.402. The general solution of the differential equation in Corollary 2.401 is  Z  −1 −1 ξ (x) = [1 − G (x)] − g (x) (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.202. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  1−α n  (βx)c o c c c e−2(βx) e(βx) − 1 exp − 1−e and q2 (x) = q1 (x) e−(βx) , x > 0. The random 1−p variable X has pd f (1.1.1236) if and only if the function ξ defined in Theorem 2.1 is of the form c 1 ξ (x) = e−(βx) , 2

x > 0.

Corollary 2.403. Let X : Ω → (0, θ) be a continuous random variable and let q1 (x) be as in Proposition 2.202. The random variable X has pd f (1.1.1236) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = cβc xc−1 , ξ (x) q1 (x) − q2 (x)

x > 0.

Corollary 2.404. The general solution of the differential equation in Corollary 2.403 is   Z −1 c c−1 −(βx)c (βx)c ξ (x) = e − cβ x e (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.203. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  −1 θ/x (1 + θ) eθ/x + 2θx−1 − 1 e and q2 (x) = q1 (x) e−θ/x , x > 0. The random variable X has pd f (1.1.1260) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

o 1n 1 + e−θ/x , 2

x > 0.

Corollary 2.405. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.203. The random variable X has pd f (1.1.1260) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) θx−2 e−θ/x = , ξ (x) q1 (x) − q2 (x) 1 − e−θ/x

x > 0.

G. G. Hamedani

380

Corollary 2.406. The general solution of the differential equation in Corollary 2.405 is  n o Z ξ (x) = 1 − e−θ/x − θx−2 e−θ/x (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

Proposition 2.204. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) =
−1 1 − xαλ and q2 (x) = q1 (x) xλ , 0 < x < 1. The random variable X has pd f (1.1.1268) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

 1 1 − xαλ , 2

0 < x < 1.

Corollary 2.407. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.204. The random variable X has pd f (1.1.1268) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) λxλ−1 = , ξ (x) q1 (x) − q2 (x) 1 − xλ

0 < x < 1.

Corollary 2.408. The general solution of the differential equation in Corollary 2.407 is   −1  Z −1 λ λ−1 ξ (x) = 1 − x − λx (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.205. Let X : Ω → (−1, 1) be a continuous random variable and let q1 (x) = sin(πx) and q2 (x) = q1 (x) (α + λ cos (πx))2 , −1 < x < 1. The random variable X has pd f (1.1.1270) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

o 1n (α + λ cos (πx))2 + (α − λ)2 , − 1 < x < 1. 2

Corollary 2.409. Let X : Ω → (−1, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.205. The random variable X has pd f (1.1.1270) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation λπ sin(πx) (α + λ cos (πx)) ξ0 (x) q1 (x) = 2 ξ (x) q1 (x) − q2 (x) (α + λ cos (πx))2 − (α − λ)2

− 1 < x < 1.

Corollary 2.410. The general solution of the differential equation in Corollary 2.409 is n o−1 ξ (x) = (α + λ cos (πx))2 − (α − λ)2 ×  Z  λπ −1 − sin(πx) (α + λ cos (πx)) (q1 (x)) q2 (x) dx + D , 2 where D is a constant.

Characterizations of Distributions

381

Proposition 2.206. Let X : Ω → R be a continuous random variable and let q1 (x) = o−1 n  b and q2 (x) = q1 (x) [1 − G (x)a ], x ∈ R. The random variable X sin π2 [1 − G (x)a ]

has pd f (1.1.1272) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

b [1 − G (x)a ] , x ∈ R. b+1

Corollary 2.411. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.206. The random variable X has pd f (1.1.1272) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) abg (x) G (x)a−1 = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x)a Corollary 2.412. The general solution of the differential equation in Corollary 2.411 is  Z  a −1 a−1 −1 ξ (x) = [1 − G (x) ] − abg (x) G (x) (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.207. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =  −1 3 3 1 − e− 2 x and q2 (x) = q1 (x) e− 2 x , x > 0. The random variable X has pd f (1.1.1276) if and only if the function ξ defined in Theorem 2.1 is of the form 1 3 ξ (x) = e− 2 x , x > 0. 2 Corollary 2.413. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.207. The random variable X has pd f (1.1.1276) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) 3 = , x > 0. ξ (x) q1 (x) − q2 (x) 2 Corollary 2.414. The general solution of the differential equation in Corollary 2.413 is  Z  3 3 −3x −1 x 2 2 ξ (x) = e − e (q1 (x)) q2 (x) dx + D , 2 where D is a constant. Proposition 2.208. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = hR n 2 2 o i−1 −x
∞ x u e 3 2 −x , x > 0. The random 0 u + u exp − b2 − cu du x and q2 (x) = q1 (x) e variable X has pd f (1.1.1282) if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−x , x > 0. 2

G. G. Hamedani

382

Corollary 2.415. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.208. The random variable X has pd f (1.1.1282) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = 1, x > 0. ξ (x) q1 (x) − q2 (x)

Corollary 2.416. The general solution of the differential equation in Corollary 2.415 is  Z  −1 x −x ξ (x) = e − e (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.209. Let X : Ω → R be a continuous random variable and let q1 (x) = [1+G(x;η)α ] and q2 (x) = q1 (x) G (x; η)α , x ∈ R. The random variable X has pd f [β+(β−α)G(x;η)α ] (1.1.1288) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

o 1n 1 + G (x; η)β , x ∈ R. 2

Corollary 2.417. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.209. The random variable X has pd f (1.1.1288) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) 2g (x; η) G (x; η)β−1 = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − G (x; η)β

Corollary 2.418. The general solution of the differential equation in Corollary 2.417 is  n o−1  Z ξ (x) = 1 − G (x; η)β − 2g (x; η) G (x; η)β−1 (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

Proposition 2.210. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) = (
−a )−1 abx−a−1 1+ xβ e−bx 
−1 xβ−1 and q2 (x) = q1 (x) 1 + xβ , x > 0. The random β−1 −bx−a +αβx 1−e variable X has pd f (1.1.1292) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

−1 α  1 + xβ , x > 0. α+1

Corollary 2.419. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.210. The random variable X has pd f (1.1.1292) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) αβxβ−1 = , x > 0. ξ (x) q1 (x) − q2 (x) 1 + xβ

Characterizations of Distributions

383

Corollary 2.420. The general solution of the differential equation in Corollary 2.419 is   −1  Z ξ (x) = 1 + xβ − αβxβ−1 (q1 (x))−1 q2 (x) dx + D ,

where D is a constant.

Proposition 2.211. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =   6 −1 and q2 (x) = q1 (x) e−βx , x > 0. The random variable X has pd f (1.1.1298) α + βx6 if and only if the function ξ defined in Theorem 2.1 is of the form 1 ξ (x) = e−βx , x > 0. 2 Corollary 2.421. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.211. The random variable X has pd f (1.1.1298) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) = β, x > 0. ξ (x) q1 (x) − q2 (x) Corollary 2.422. The general solution of the differential equation in Corollary 2.421 is  Z  −1 βx −βx ξ (x) = e − βe (q1 (x)) q2 (x) dx + D , where D is a constant. Proposition 2.212. Let X : Ω → R be a continuous random variable and let q1 (x) =

φ(λx) f0 (x)

and q2 (x) = q1 (x) Φ (λx)2 , x ∈ R. The random variable X has pd f (1.1.1304) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

o 1n 1 + Φ (λx)2 , x ∈ R. 2

Corollary 2.423. Let X : Ω → R be a continuous random variable and let q1 (x) be as in Proposition 2.212. The random variable X has pd f (1.1.1304) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) λφ (λx) Φ (λx) = , x ∈ R. ξ (x) q1 (x) − q2 (x) 1 − Φ (λx)2 Corollary 2.424. The general solution of the differential equation in Corollary 2.423 is  n o−1  Z 2 −1 ξ (x) = 1 − Φ (λx) − λφ (λx) Φ (λx) (q1 (x)) q2 (x) dx + D ,

where D is a constant.

G. G. Hamedani

384

Proposition 2.213. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) =   β n o α+ β2 eαx− x [1+xc]k x αx− βx   and q (x) = q (x) exp −e , x > 0. The random variable 2 1 β β −1 α+

x2

eαx− x +ckxc−1 [1+xc]

X has pd f (1.1.1334) if and only if the function ξ defined in Theorem 2.1 is of the form ξ (x) =

n o β 1 exp −eαx− x , x > 0. 2

Corollary 2.425. Let X : Ω → (0, ∞) be a continuous random variable and let q1 (x) be as in Proposition 2.213. The random variable X has pd f (1.1.1334) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation   β β ξ0 (x) q1 (x) = α + 2 eαx− x , x > 0. ξ (x) q1 (x) − q2 (x) x

Corollary 2.426. The general solution of the differential equation in Corollary 2.425 is   n o Z  n o β −1 αx− βx αx− βx αx− βx ξ (x) = exp e − α+ 2 e exp −e (q1 (x)) q2 (x) dx + D , x where D is a constant.

Proposition 2.214. Let X : Ω → (0,

1) be a continuous random variable and let q1 (x) ≡ −1 x 1 and q2 (x) = Φ sinh log 1−x , 0 < x < 1. The random variable X has pd f (1.1.1338) if and only if the function ξ defined in Theorem 2.1 is of the form      1 x −1 ξ (x) = 1 + Φ sinh log , 0 < x < 1. 2 1−x

Corollary 2.427. Let X : Ω → (0, 1) be a continuous random variable and let q1 (x) be as in Proposition 2.214. The random variable X has pd f (1.1.1338) if and only if there exist functions q2 and ξ defined in Theorem 2.1 satisfying the following differential equation ξ0 (x) q1 (x) ξ (x) q1 (x) − q2 (x) =−

q 1/2 φ x x(1−x) 1+[log( 1−x )]



x sinh−1 log 1−x 

, 0 < x < 1. x 1 − Φ sinh−1 log 1−x

Corollary 2.428. The general solution of the differential equation in Corollary 2.427 is     −1 x −1 ξ (x) = 1 − Φ sinh log × 1−x   

−1 x q 1/2 φ sinh log Z 1−x x  x(1−x) 1+[log( 1−x  )]  

q2 (x) dx + D −1  x , 1 − Φ sinh log 1−x where D is a constant.

Characterizations of Distributions

385

2.2 Characterization in Terms of Hazard Function The hazard function, hF , of a twice differentiable distribution function, F, satisfies the following first order differential equation f 0 (x) h0F (x) = − hF (x). f (x) hF (x) It should be mentioned that for many univariate continuous distributions, the above equation is the only differential equation available in terms of the hazard function. In this subsection we present non-trivial characterizations of some of the new distributions in terms of the hazard function, which are not of the above trivial form. Proposition 2.215. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.2), for γ = 1, if and only if its hazard function hF (x) satisfies the following differential equation ( ) xβ−1 (β + λx) 0 −β λx d , x > 0. hF (x) − λhF (x) = αθ e
dx 1 + x β eλx θ

Proof. If X has pdf (1.1.2), then clearly the above differential equation holds. If the differential equation holds, then ( ) o β−1 (β + λx) d x d n −λx e hF (x) = αθ−β , x > 0, dx dx 1 + x
β eλx θ

from which we arrive at the hazard function corresponding to the pdf (1.1.2), for γ = 1.



Remark 2.2. For β = 1, we have a much simpler differential equation
λθ2 − θeλx 1 + λ + λ2 x 0 hF (x) − λhF (x) = , x > 0.
2 θ + xeλx

A Proposition similar to that of Proposition 2.215 will be stated (without proof) for ACLW, BType II, OLEW, OBIIIGNB, LSCp, BH-G, P-EHR, RNMW, EW-G, MOEFW, LKu, AOLLL-G, APP, ADFIF, KwTIHL-G, CE-N{GW}, Γ2 -W, GOLi-G, LDSWeibull, unit-ISDL, TDLW, RC, BHE, TNWP, LOGOMA, ETF, EPL-G, EW-EW, WGBXII, EG-PL (for b = 1), GK-G, EGuE, LTL, Go-LBE, TIIPTL-G, GMEPS, MOW-G, WMOL, KR, LWP, IGC, INBC, IBC, IPC, PM, WAPIE, E-Epsilon (for α = 1), GAPIE, G-EGE, PLG, EGSGu (for b = 1), unit-Lindley, ELBE (for β = 1), ETIW-G (for a = 1), NET, EEW (for α = 1), TLMO-G (for b = 1), MOEPL, erf-G, TIITLPI-G, GOLE, WNB, NE-F, OFIL distributions. Proposition 2.216. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.46) if and only if its hazard function hF (x) satisfies the following differential equation   h0 (x; η) d 1 + βH (x; η) 2 0 hF (x) − hF (x) = θ h (x; η) , x > 0. h (x; η) dx β + θ (1 + βH (x; η))

G. G. Hamedani

386

Proposition 2.217. Let X : Ω → (1, ∞) be a continuous random variable. The random variable X has pdf (1.1.52) if and only if its hazard function hF (x) satisfies the following differential equation h0F (x) + x−1 hF (x) = abxb−2 , x > 1. Proposition 2.218. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.88) if and only if its hazard function hF (x) satisfies the following differential equation β h iα−1 (α − 1) βxβ−1 e−x 2 −xβ h (x) = a αβ 1 − e × F −xβ  1−e    β  d  xβ−1 e−x , x > 0. n o  n o   2    dx  β α  1 − 1 − e−xβ α  a + 1 − 1 − e−x

h0F (x) −

Proposition 2.219. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.162) if and only if its hazard function hF (x) satisfies the following differential equation b0c,k,η (x)

d h0F (x) − hF (x) = sβbc,k,η (x) bc,k,η (x) dx

(

{1 − β [1 − Bc,k,η (x)]}−s−1 {1 − β [1 − Bc,k,η (x)]}−s − 1

)

, x > 0.

Proposition 2.220. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.206) if and only if its hazard function hF (x) satisfies the following differential equation   1 τν −1 d cosh(logx) 0 hF (x) + hF (x) = x , x > 0. x π dx ν2 sinh2 (logx) + 1 Proposition 2.221. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.206) if and only if its hazard function hF (x) satisfies the following differential equation g0 (x; η) d h0F (x) − hF (x) = g (x; η) g (x; η) dx

(    ) θ 1 − log G(x; η) + 1    , x ∈ R. G (x; η) 1 − log G (x; η)

Proposition 2.222. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.234) if and only if its hazard function hF (x) satisfies the following differential equation xh0F (x) − khF (x) = b (cx − k) ecx , x > 0.

Characterizations of Distributions

387

Proposition 2.223. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.250) if and only if its hazard function hF (x) satisfies the following differential equation xh0F (x) +

1 λβ hF (x) = √ (3 + 2λx) eλx , x > 0. 2x 2 x

Proposition 2.224. Let X : Ω → R be a continuous random variable. The random variable X has pdf (1.1.252) if and only if its hazard function hF (x) satisfies the following differential equation

xh0F (x) −

g0 (x; η) hF (x) = αg (x; η) × g (x; η) ( ) β−2  λG (x; η) + βG (x; η) d G (x; η) , x ∈ R. dx G (x; η)β+1

Proposition 2.225. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.298) if and only if its hazard function hF (x) satisfies the following differential equation
h0F (x) − α − β/x2 hF (x) = b exp [αx + β/x] × ( ) d α + β/x   , x > 0. dx 1 − (1 − a) exp −bκαβ (x)

Proposition 2.226. Let X : Ω → R be a continuous random variable. The random variable X has pdf (1.1.330) if and only if its hazard function hF (x) satisfies the following differential equation (β − 1) g (x; η) hF (x) = βγ (eα − 1) G (x; η)β−1 × G (x; η)  γ−1 ( β g (x; η) 1 − G (x; η) d h h n  γ o ii × dx log eα − 1 − 1 − G (x; η)β (eα − 1) ) 1 h n  γ o i , x ∈ R. eα − 1 − 1 − G (x; η)β (eα − 1)

h0F (x) −

Proposition 2.227. Let X : Ω → R be a continuous random variable. The random variable X has pdf (1.1.344) if and only if its hazard function hF (x) satisfies the following differential equation

G. G. Hamedani

388

x ∈ R.

(α − 1) g (x; φ) h0F (x) + hF (x) = −αβG (x; φ)α−1 × G (x; φ) ) ( g (x; φ)G (x; φ)α−1 d   ,   dx − log(1 − β) G (x; φ)α + (1 − β) G (x; φ)α G (x; φ)α + G (x; φ)α

Proposition 2.228. Let X : Ω → [1, ∞) be a continuous random variable. The random variable X has pdf (1.1.366) if and only if its hazard function hF (x) satisfies the following differential equation h0F (x) +

−β β+1 hF (x) = −β2 (logα)2 αx x−2(β+1) , x > 1. x

Proposition 2.229. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.380) if and only if its hazard function hF (x) satisfies the following differential equation h0F (x) −

λ

hF (x) (λ2 + x − 2λ) (λ2 + x − λ)

= 0, x > 0.

Proposition 2.230. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.440) if and only if its hazard function hF (x) satisfies the following differential equation   h ia−1   λ−1 λ    G (x) 1 − G (x) g0 (x) d  0   hF (x) − hF (x) = 2abλg (x) h ia+1 h ia  , x ∈ R. g (x) dx  1−G(x)λ λ     1 + G (x) 1− λ 1+G(x)

Proposition 2.231. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.488) if and only if its hazard function hF (x) satisfies the following differential equation h0F (x) −

φ0 (x) hF (x) = (λ + 1) φ (x)2 (1 − Φ (x))−(λ+2) , x > 0. φ (x)

Proposition 2.232. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.500) if and only if its hazard function hF (x) satisfies the following differential equation  h  ia−1    β−1 −αxβ   x 1 − log 1 − e 0 β−1 −αxβ d hF (x) + αβx hF (x) = αβe

, x > 0.  dx  γ a, − log 1 − e−αxβ  

Characterizations of Distributions

389

Proposition 2.233. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.520), for α = 1, if and only if its hazard function hF (x) satisfies the following differential equation

h0F (x) −


n 4
−1 o 3x2 5 3 d 2 2 h (x) = θ θ + x θ + 6 + θx θ x + 3θx + 6 , x > 0. F θ + x3 dx

Proposition 2.234. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.522) if and only if its hazard function hF (x) satisfies the following differential equation h0F (x) −

(α − 1) g (x; ϕ) hF (x) = αλ2 G (x; ϕ)α−1 × G (x; ϕ) ( ) d g (x; ϕ)  2   , x ∈ R. dx 1 − G (x; ϕ)α 1 + λ − G (x; ϕ)α

Proposition 2.235. Let X : Ω → (θ, ∞) be a continuous random variable. The random variable X has pd f (1.1.540) if and only if its hazard function hF (x) satisfies the following differential equation 2 γ (γ − 1) (x − θ)γ−2 , x > θ. h0F (x) + hF (x) = x τx Proposition 2.236. Let X : Ω → (0, 1) be a continuous random variable. The random variable X has pd f (1.1.570) if and only if its hazard function hF (x) satisfies the following differential equation h0F (x) − 2 (1 − x)−1 hF (x) = 
 −2λ3 (1 − x)−2 1 − λ − 2λ2 x + λ2 + λ − 1 [1 + (1 − λ − 2λ2 ) x2 + 2(λ2 + λ − 1)x]

2

, 0 < x < 1.

Proposition 2.237. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.576) if and only if its hazard function hF (x) satisfies the following differential equation β

h0F (x) + βλβ xβ−1 hF (x) = ανβλβ (1 − p) e−(λx) ×   h i   β α−1    xβ−1 1 − e−(λx) d    h   i    ν  , x > 0. dx  β α β α    p + (1 − p) 1 − e−(λx)  1 − p + (1 − p) 1 − e−(λx)

Proposition 2.238. Let X : Ω → (−1, 1) be a continuous random variable. The random variable X has pd f (1.1.578) if and only if its hazard function hF (x) satisfies the following differential equation

G. G. Hamedani

390

π sin (πx) hF (x) = h0F (x) + 1 + cos (πx)

1 + cos (πx) 1 − x − π1 sin(πx)

!

, − 1 < x < 1.

Proposition 2.239. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.604) if and only if its hazard function hF (x) satisfies the following differential equation h0F (x) −

β−1 δ2 β2 λ (λ − 1) 2(β−1) −δ( x )β hF (x) = x e θ , x > 0. x θβ+1

Proposition 2.240. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.622) if and only if its hazard function hF (x) satisfies the following differential equation   −θx− αβ (eβx−1) × h0F (x) − θ + αeβx hF (x) = ae  
  βx θ + αe d h  i n h  io , x > 0. α α −θx− β (eβx−1)  dx  1 − 1 − e−θx− β (eβx −1) b − log 1 − 1 − e

Proposition 2.241. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.624) if and only if its hazard function hF (x) satisfies the following differential equation α

h0F (x) − αβx−α−1 hF (x) = θαβαe−(β/x) ×     −α−1 −(β/x)α   1 + λ − 2λe x d , x > 0. α α
 dx  1 − e−(β/x) 1 + λ 1 − e−(β/x) 

Proposition 2.242. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.638) if and only if its hazard function hF (x) satisfies the following differential equation

h0F (x) −

g0 (x; φ) hF (x) = αθ2 g (x; φ) g (x; φ) 
α−1 n
α o    − logG (x; φ) 1 + − log G(x; φ) d n h io , x ∈ R.  dx  G (x; φ) 1 + θ 1 + − log G (x; φ)
α

Proposition 2.243. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.646) if and only if its hazard function hF (x) satisfies the following differential equation

Characterizations of Distributions

h0F (x) +

391

ac αθ −( x )a a  x a−1 e b hF (x) = b b b β    h  a iα−1 
x    x a − c ln 1 − e−( b )   d  b β , x > 0.    α a −( bx )  dx  h c   ai − ln 1−e x    1 − e−( b ) e β 

Proposition 2.244. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.650) if and only if its hazard function hF (x) satisfies the following differential equation       (1 + xa )bθ−1 a−1 0 a−1 d hF (x) = βθabx , x > 0. hF (x) − h i −β+1  x dx   (1 + xa )bθ − 1 

Proposition 2.245. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.658) if and only if its hazard function hF (x) satisfies the following differential equation ( ) 2 β−1 λxβ αβλ β d x e h0F (x) + λβx−1 hF (x) = e−λx , x > 0. 1+λ dx 1 + λxβ 1+λ

Proposition 2.246. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.662) if and only if its hazard function hF (x) satisfies the following differential equation g0 (x; ϕ) d h0F (x) − hF (x) = αabg (x; ϕ) g (x; ϕ) dx

(

b−1

G (x; ϕ)a−1 [1 − αG (x; ϕ)a ] b

[1 − αG (x; ϕ)a ] − (1 − α)b

)

, x ∈ R.

Proposition 2.247. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.672) if and only if its hazard function hF (x) satisfies the following differential equation   n
− σ1 −1
− σ1 o   θx θx  exp −B e − 1 αθB θx d  e − 1 n o h0F (x) − θhF (x) = e , x > 0. 1 −   σ dx   1 − exp −B (eθx − 1) σ

Proposition 2.248. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.696) if and only if its hazard function hF (x) satisfies the following differential equation

G. G. Hamedani

392

x > 0.

h0F (x) + βλβ (λ + x)−(β+1) hF (x)         λ β d (λ + x)−(β+1) β 1−( λ+x )     , = log (α) βλ α β λ λ β  dx    1 + α − α1−( λ+x ) log 1 + α − α1−( λ+x )  

Proposition 2.249. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.700) if and only if its hazard function hF (x) satisfies the following differential equation      1 θe−x/β d  x 0 hF (x) + hF (x) = , x > 0. h  i γ+1  β β2 dx   1 + x e−x/β  β

Proposition 2.250. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.704) if and only if its hazard function hF (x) satisfies the following differential equation

h0F (x) −

g0 (x; η) hF (x) = 2αβg (x; η) g (x; η)     β d 1 − [1 − G (x; η)] n o , x ∈ R. dx  [1 − G (x; η)] 2 − [1 − G (x; η)]β 

Proposition 2.251. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.730) if and only if its hazard function hF (x) satisfies the following differential equation   d M 0 (θH (x)) g0 (x) 0 hF (x) − hF (x) = αg (x) , x > 0. g (x) dx M (θH (x)) Proposition 2.252. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.756) if and only if its hazard function hF (x) satisfies the following differential equation

h0F (x) −

g0 (x; ζ) hF (x) = λθg (x; ζ) g (x; ζ) i  h  2  1 + G (x; ζ) /2 G (x; ζ)θ−1  d , x ∈ R. dx  {1 − G (x; ζ) [1 + G (x; ζ)] /2}θ+1 

Characterizations of Distributions

393

Proposition 2.253. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.762) if and only if its hazard function hF (x) satisfies the following differential equation βa2 (1 + a) eax α (   ) d (1 + x) (1 + a) eax α −1 − , x ∈ R. dx (1 + a + ax)2 α (1 + a + ax) α

h0F (x) − ahF (x) =

Proposition 2.254. Let X : Ω → (a, b) be a continuous random variable. The random variable X has pd f (1.1.770) if and only if its hazard function hF (x) satisfies the following differential equation

h0F (x) +

( )
−α   ln ax 1 b −α −1 d , a < x < b. hF (x) = αβ ln x

x a dx 1 − ln b −α ln x α a a

Proposition 2.255. Let X : Ω → (β, ∞) be a continuous random variable. The random variable X has pd f (1.1.772) if and only if its hazard function hF (x) satisfies the following differential equation 1 h0F (x) + hF (x) = αβ2−αx−2 (1 + α log x)−2 , x > β. x Proposition 2.256. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.778) if and only if its hazard function hF (x) satisfies the following differential equation g0 (x) d h0F (x) − hF (x) = g (x) g (x) dx

(

2

1 − ρθ2 G (x) 

 G(x) 1 − ρθG (x) 1 − θG (x)

)

, x ∈ R.

Proposition 2.257. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.780) if and only if its hazard function hF (x) satisfies the following differential equation

h0F (x) −

g0 (x) hF (x) = rθg (x) × g (x) h  
r+1 i   1 + ρ − 2ρθG (x) − ρ 1 − θG (x) d


r  , x ∈ R. dx  1 − ρθG (x) 1 − θG (x) 1 − 1 − θG (x) 

Proposition 2.258. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.782) if and only if its hazard function hF (x) satisfies the following differential equation

G. G. Hamedani

394

h0F (x) −

g0 (x) hF (x) = mθg (x) × g (x)    m−1    ρθGx + 1 2ρθθG(x) + ρ + 1 − ρ  d  
m  , x ∈ R.  dx  ρθGx + 1 θG(x) + 1 − 1

Proposition 2.259. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.784) if and only if its hazard function hF (x) satisfies the following differential equation ) ( g0 (x) d (1 + ρ) eλG(x) − ρ 0 hF (x) − hF (x) = λg (x) , x ∈ R. g (x) dx eλG(x) − ρ Proposition 2.260. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.796) if and only if its hazard function hF (x) satisfies the following differential equation h0F (x) −

γ−1 γ2 x2(γ−1) (x/β)γ hF (x) = e , x > 0. x β2γ

Proposition 2.261. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.800) if and only if its hazard function hF (x) satisfies the following differential equation

h0F (x) −

  β−1    e−λ/x −1 −λ/x   α e   α  α−1

λ ϕβλ ln(α)e−λ/x d , x > 0. h (x) = F h i −λ/x β+1  x2 α−1 dx    α−αe 2     x α−1

Proposition 2.262. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.802) if and only if its hazard function hF (x) satisfies the following differential equation h0F (x) −

2x hF (x) , 0 < x < d. d 2 − x2

Proposition 2.263. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.804) if and only if its hazard function hF (x) satisfies the following differential equation c log α −c/x αc logα d h0F (x) − e hF (x) = 2 x α − 1 dx

(

e−c/x x2



α−1 −c/x α − αe

)

, x > 0.

Characterizations of Distributions

395

Proposition 2.264. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.806) if and only if its hazard function hF (x) satisfies the following differential equation

h0F (x) + hF (x) = θγ βγ − (β − 1)

γ
λ

d e−x dx

(

(β − e−x )

γ−1

βγ − (β − e−x )γ


λ+1

)

, x > 0.

Proposition 2.265. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.858) if and only if its hazard function hF (x) satisfies the following differential equation

h0F (x) −

αxα−1 hF (x) = αβ2 (1 + xα ) × 1 + xα  d 

α−1

 

x h h  ii , x > 0. dx  (β + 1 + βxα ) 1 − θ β+1+βxα e−βxα  β+1

Proposition 2.266. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.884) if and only if its hazard function hF (x) satisfies the following differential equation h0F (x) + hF (x) = ae−2x+e

−x

 −x −2 ee − 1 , x ∈ R.

Proposition 2.267. Let X : Ω → (0, 1) be a continuous random variable. The random variable X has pd f (1.1.886) if and only if its hazard function hF (x) satisfies the following differential equation h0F (x) −

 2 2 θ hF (x) = − , 0 < x < 1. 1−x (1 − x) (x − 1 − θ)

Proposition 2.268. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.938), for a = 1, if and only if its hazard function hF (x) satisfies the following differential equation   2 αθ−2 1 + θx + θx2 e−2x/θ , x > 0. h0F (x) + θ−1 hF (x) = 
2 1 + θx e−x/θ Proposition 2.269. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.942), for β = 1, if and only if its hazard function hF (x) satisfies the following differential equation h0F (x) −

g0 (x; η) b (b + 1) g (x; η)2 hF (x) = , x ∈ R. g (x; η) (1 − G (x; η))b+2

G. G. Hamedani

396

Proposition 2.270. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.948) if and only if its hazard function hF (x) satisfies the following differential equation αβ sin(βx) hF (x) = 1 − α cos (βx)
αβλ2 β2 + λ2 (1 − α cos (βx))(λ sin(βx) + β cos (βx))

h0F (x) − −

[β2 + λ2 − αλ2 cos(βx) + αβλ sin(βx)]2

, x > 0.

Proposition 2.271. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.992) if and only if its hazard function hF (x) satisfies the following differential equation h0F (x) −

βλ hF (x) , x > 0. 1 − βλx

Proposition 2.272. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.1016), for b = 1, if and only if its hazard function hF (x) satisfies the following differential equation

h0F (x) −

  i−1  g0 (x; η) d h hF (x) = 2g (x; η) G (x; η) 1 − δG (x; η) , x ∈ R. g (x; η) dx

Proposition 2.273. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.1056) if and only if its hazard function hF (x) satisfies the following differential equation  h
β iγ−1   x   1+ λ  β−1 0 −β β−1 d hF (x) − hF (x) = βγλ x h i , x > 0. γ  1 + x
β − α   x dx  λ

Proposition 2.274. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.1066) if and only if its hazard function hF (x) satisfies the following differential equation     2  G(x;θ)     exp −   0 G(x;θ) g (x; θ) 2g (x; θ) d 0 n h io , x ∈ R. hF (x) − hF (x) = √ 2 g (x; θ) π dx   G (x; θ) 1 − erf G(x;θ)     G(x;θ)

Proposition 2.275. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.1086) if and only if its hazard function hF (x) satisfies the following differential equation

Characterizations of Distributions

h0F (x) + αθxα−1 hF (x) =

397

2αβθ3 −θxα e × 3 +2 θ   
 α α α   xα−1 θ + x2α 1 − θx (θx +2) e−θx  

d dx  

θ3 +2

h



1− 1− 1+

θxα (θxα +2) θ3 +2

i2

e−2θx

α

, x

> 0.

 

Proposition 2.276. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.1106) if and only if its hazard function hF (x) satisfies the following differential equation h0F (x) −

(c − 1) g (x; φ) hF (x) = cG (x; φ)c−1 × G (x; φ) ( ) d g (x; φ) (a + (b − a) G (x; φ)c ) , x 3 dx (1 − G (x; φ)c )

∈ R.

Proposition 2.277. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.1118) if and only if its hazard function hF (x) satisfies the following differential equation β

h0F (x) −

β−1 β2 α2β kpx2(β−1)e−(xα) hF (x) = −  , x > 0.  β 2 x −(xα) 1 − pe

Proposition 2.278. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.1138) if and only if its hazard function hF (x) satisfies the following differential equation g0 (x; η) d hF (x) = g (x; η) h0F (x) − g (x; η) dx

(

 ) eθG(x;η) 1 − θG (x; η) 1 − G (x; η) eθG(x;η)

, x ∈ R.

Proposition 2.279. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.1196) if and only if its hazard function hF (x) satisfies the following differential equation  α−1 h α iθ−1  β β     1 + 1 + − 1 x x 2 αβθ d 0 hF (x) + hF (x) = 2 , x > 0. h α iθ  x x dx   1+ βx −1  e −1

2.3 Characterization in Terms of the Reverse (or Reversed) Hazard Function The reverse hazard function, rF , of a twice differentiable distribution function, F , is defined as

G. G. Hamedani

398

rF (x) =

f (x) , x ∈ support o f F. F (x)

In this subsection we present characterizations of some of the new distributions in terms of the reverse hazard function. Proposition 2.280. Let X : Ω → R be a continuous random variable. The random variable X has pdf (1.1.6) if and only if its reverse hazard function rF (x) satisfies the following differential equation (β + 1) g (x; η) rF (x) = G (x; η)  β−2  0 βθβ [G (x; η)]−β−1 G (x; η) g (x; η) G (x; η) − (β − 1) [g (x; η)] , x ∈ R,

rF0 (x) +

with boundary condition limx→∞ rF (x) = 0 for β > 1.

Proof. If X has pdf (1.1.6), then clearly the above differential equation holds. If the differential equation holds, then o β−1 o  d n d n , x ∈ R, [G (x; η)]β+1 rF (x) = βθβ g (x; η) G (x; η) dx dx from which we arrive at the hazard function corresponding to the pd f (1.1.6).



A Proposition similar to that of Proposition 2.280 will be stated (without proof) for EWE, IWE, MG, ETPF, GLL, EEL, HS-R, GaL, OHC-KG, APP, MFr, MOAPIE, GG, TLSS, THL, L-Logistic, TTIGL, TDLW, RC, TIK-G, GIL, EW-EW (for θ = 1), EGuE (for α = 1), DTEVTI, IGC, E-Epsilon, EOLL-G, EGSGu, UW, ELBE, ECTE, ETIW-G, TB-G, GOIE-G, EEW, TLMO-G, NUL, HCW, OFIL, ESN, SK-G, WE, RE-G distributions. Proposition 2.281. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.14) if and only if its reverse hazard function rF (x) satisfies the following differential equation ( ) 1 − e−αλx 0 −λx d
, x > 0, rF (x) + λrF (x) = θλe 1 dx 1 − e−λx − 1+α 1 − e−λ(1+α)x with boundary condition limx→∞ rF (x) = 0.

Proposition 2.282. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.18) if and only if its reverse hazard function rF (x) satisfies the following differential equation αβ

rF0 (x) +

2 α2 (α + 1) β2 e− x rF (x) = −   , x > 0, αβ 2 x x4 α + 1 − e− x

with boundary condition limx→∞ rF (x) = 0.

Characterizations of Distributions

399

Proposition 2.283. Let X : Ω → R be a continuous random variable. The random variable X has pdf (1.1.22) if and only if its reverse hazard function rF (x) satisfies the following differential equation

rF0 (x) −


g0 (x; β) 1 − αG (x; β)α (α + 1) rF (x) = − α2 g (x; β)2 G (x; β)−2 , x ∈ R, α+1 G (x; β) G (x; β)

with boundary condition limx→∞ rF (x) = (1 − α) limx→∞ g (x; β). Proposition 2.284. Let X : Ω → (0, β) be a continuous random variable. The random variable X has pdf (1.1.124) if and only if its reverse hazard function rF (x) satisfies the following differential equation   α    1 + γ − 2γ βx   0 −1 −1 d  α , x ∈ (0, β), rF (x) + x rF (x) = αθx  dx   1+γ−γ x  β with boundary condition limx→β− rF (x) =

αθ(1−γ) β

for γ ≤ 1.

Proposition 2.285. Let X : Ω → (0, 1) be a continuous random variable. The random variable X has pdf (1.1.222) if and only if its reverse hazard function rF (x) satisfies the following differential equation

rF0 (x) + x−1 rF (x)


 σβ (β + σ) xβ−2 θ (β + σ) − σ 1 + xβ+1 =− , x ∈ (0, 1),  2 θ (β + σ) − σxβ

with boundary condition limx→0+ rF (x) = 0, for δ > 1.

Proposition 2.286. Let X : Ω → R be a continuous random variable. The random variable X has pdf (1.1.248) if and only if its reverse hazard function rF (x) satisfies the following differential equation 1 d rF0 (x) − rF (x) = αλex/θ θ dx with boundary condition limx→−∞ rF (x) = 0.

(

1 + ex/θ


−λ )

1 − 1 + ex/θ


−λ

, x ∈ R,

Proposition 2.287. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.266) if and only if its reverse hazard function rF (x) satisfies the following differential equation      −σx2   x sinh α 1 − e 2 d

rF0 (x) + 2σxrF (x) = ασe−σx , x > 0, dx  cosh α 1 − e−σx2 − 1  with boundary condition limx→0 rF (x) = 0.

400

G. G. Hamedani

Proposition 2.288. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.280) if and only if its reverse hazard function rF (x) satisfies the following differential equation   1+α+β+x 0 −1 2 −1 d , x > 0, rF (x) + x rF (x) = αβ x dx (β + x) [x (1 + β) + (1 + α + β) β] with boundary condition limx→∞ rF (x) = 0. Proposition 2.289. Let X : Ω → R be a continuous random variable. The random variable X has pdf (1.1.312) if and only if its reverse hazard function rF (x) satisfies the following differential equation        G(x) G(x)   k coth aK 0 1−G(x) 1−G(x) d g (x) rF0 (x) − rF (x) = ag (x) , x ∈ R,  g (x) dx  (1 − G (x))2

with boundary condition limx→−∞ rF (x) = ak (0)coth (aK (0)) limx→−∞ g (x) .

Proposition 2.290. Let X : Ω → [1, ∞) be a continuous random variable. The random variable X has pdf (1.1.366) if and only if its reverse hazard function rF (x) satisfies the following differential equation ( ) −β d β + 1 1−x α rF (x) = β (logα)x−(β+1) , x > 1, rF0 (x) + x dx α1−x−β − 1 with boundary condition limx→∞ rF (x) = 0. Proposition 2.291. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.464) if and only if its reverse hazard function rF (x) satisfies the following differential equation rF0 (x) + λrF (x) = −βαβ x−β−2 (β + 1 = λx) e−λx , x > 0, with boundary condition limx→∞ rF (x) = 0. Proposition 2.292. Let X : Ω → R be a continuous random variable. The random variable X has pdf (1.1.478) if and only if its reverse hazard function rF (x) satisfies the following differential equation rF0 (x) + log (α)g (x) rF (x) = (α − 1) θαG(x) × ( ) d g (x)

 , x dx αG(x) − 1 (α − 1) θ − (1 − θ) αG(x) − 1

with boundary condition limx→∞ rF (x) =

αθ log(α) α−1

limx→∞ g (x) .

∈ R,

Characterizations of Distributions

401

Proposition 2.293. Let X : Ω → R be a continuous random variable. The random variable X has pdf (1.1.482) if and only if its reverse hazard function rF (x) satisfies the following differential equation h   i−1  0 βx d 2βx rF (x) − βrF (x) = βe 1+e arctan eβx , x ∈ R, dx

with boundary condition limx→∞ rF (x) = 0.

Proposition 2.294. Let X : Ω → R be a continuous random variable. The random variable X has pdf (1.1.486) if and only if its reverse hazard function rF (x) satisfies the following differential equation ) ( g (x) 1 − e−λG(x) 0 −λG(x) d , x ∈ R, rF (x) + λg (x) rF (x) = 2λe dx 1 + e−λG(x) with boundary condition limx→∞ rF (x) = λ limx→−∞ g (x). Proposition 2.295. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.512) if and only if its reverse hazard function rF (x) satisfies the following differential equation   (1 − λ) ex + 3λ + 1 0 x d rF (x) − rF (x) = 2e , x > 0, dx (e2x − 1) (1 + 2λ + ex ) with boundary condition limx→∞ rF (x) = 0.

Proposition 2.296. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.520) if and only if its reverse hazard function rF (x) satisfies the following differential equation

rF0 (x) + θrF (x) =



4 −θx

αθ e θ4 + 6



    3 d θ+x h i , x > 0, dx  1 − 1 + θx(θ2 x2 +3θx+6) e−θx  θ4 +6

with boundary condition limx→∞ rF (x) = 0.

Proposition 2.297. Let X : Ω → (0, 1) be a continuous random variable. The random variable X has pdf (1.1.542) if and only if its reverse hazard function rF (x) satisfies the following differential equation d rF0 (x) + x−1 rF (x) = bmb x−1 dx

(

(1 − x)b−1

xb (1 − m)b + mb (1 − x)b

with boundary condition limx→1 rF (x) = 0, for b > 1.

)

, 0 < x < 1,

G. G. Hamedani

402

Proposition 2.298. Let X : Ω → R be a continuous random variable. The random variable X has pdf (1.1.574) if and only if its reverse hazard function rF (x) satisfies the following differential equation     −x b (1 + λ) (1 + e ) − 2λ d h i , x ∈ R, rF0 (x) + rF (x) = be−x dx  (1 + e−x ) (1 + λ) (1 + e−x )b − λ 

with boundary condition limx→∞ rF (x) = 0.

Proposition 2.299. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.576) if and only if its reverse hazard function rF (x) satisfies the following differential equation β

rF0 (x) + βλβ xβ−1 rF (x) = ανβλβ (1 − p) e−(λx) ×   h i   β α−1    xβ−1 1 − e−(λx) d    , h   i    ν  dx  β α β α   p + (1 − p) 1 − e−(λx)  p + (1 − p) 1 − e−(λx) − pν 

x > 0, with boundary condition limx→∞ rF (x) = 0.

Proposition 2.300. Let X : Ω → (−1, 1) be a continuous random variable. The random variable X has pdf (1.1.578) if and only if its reverse hazard function rF (x) satisfies the following differential equation π sin(πx) rF0 (x) + rF (x) = − 1 + cos (πx)

1 + cos (πx) 1 + x + π1 sin(πx)

!2

,

− 1 < x < 1,

with boundary condition limx→0 rF (x) = 2. Proposition 2.301. Let X : Ω → R be a continuous random variable. The random variable X has pdf (1.1.620) if and only if its reverse hazard function rF (x) satisfies the following differential equation

rF0 (x) +

(a + 1) g (x; η) rF (x) = ab (1 + G (x; η))−a−1 × 1 + G (x; η) ( ) d g (x; η)   , x ∈ R, dx 1 − (1 + G (x; η))−a

with boundary condition limx→∞ rF (x) =

ab2−a−1 1−2−a

limx→∞ g (x; η) .

Proposition 2.302. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pdf (1.1.634) if and only if its reverse hazard function rF (x) satisfies the following differential equation

Characterizations of Distributions

rF0 (x) + (α + 1) x−1 rF (x)

2 −α−1

= αβ x

d dx

with boundary condition limx→∞ rF (x) = 0.



1 + λx−α β + λ + βλx−α

403 

, x > 0,

Proposition 2.303. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.646) if and only if, for θ = 1, its hazard function rF (x) satisfies the following differential equation

rF0 (x) +

a  x a−1 ac α −( x )a rF (x) = e b × b b b β   a iα−1
 c h −( xb )   x a   − ln 1 − e b d β h i , x > 0, x a  dx    1 − e−( b )

with boundary condition limx→∞ rF (x) = 0 for α > 1.

Proposition 2.304. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.672) if and only if, for α = 1, its hazard function rF (x) satisfies the following differential equation − 1 −2 θ2 B (1 + σ) 2θx  θx σ e e − 1 , x > 0, σ2 with boundary condition limx→∞ rF (x) = 0. rF0 (x) − θrF (x) = −

Proposition 2.305. Let X : Ω → (A, B) be a continuous random variable. The random variable X has pd f (1.1.742) if and only if its hazard function rF (x) satisfies the following differential equation rF0 (x) + rF

(x) =

e−2x e−e

−A

e−e−x − e−e−A

with boundary condition limx→B rF (x) =

−B


2 , A < x < B,

e−B e−e . e−e−B −e−e−A

Proposition 2.306. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.778) if and only if its hazard function rF (x) satisfies the following differential equation

rF0 (x) −

g0 (x) rF (x) = (1 − θ) (1 − ρθ)g (x) × g (x) ( ) d G−1 (x)

, x ∈ R, dx 1 − ρθG (x) 1 − θG (x)

with boundary condition limx→∞ rF (x) = (1 − θ) (1 − ρθ) limx→∞ g (x).

G. G. Hamedani

404

Proposition 2.307. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.802) if and only if its hazard function rF (x) satisfies the following differential equation  
d+x −λd/2   2 αλd d 2x d−X rF (x) = 2 , 0 < x < d, rF0 (x) − 2 2 d −x d − x2 dx  1 − d+x
−λd/2  d−X

with boundary condition limx→0 rF (x) = 0.

Proposition 2.308. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.808) if and only if its hazard function rF (x) satisfies the following differential equation rF0 (x) −

g0 (x; η) rF (x) = αβg (x; η) × g (x; η)  ioβ  n h G(x;η)α     G (x; η) − log d G(x;η)α +G(x;η)α   , x ∈ R, α α  dx   G (x; η) G (x; η) + G (x; η) 

with boundary condition limx→∞ rF (x) = 0.

Proposition 2.309. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.884) if and only if its hazard function rF (x) satisfies the following differential equation     −x −x a−1   −e −e   e 1−e 0 −x d
rF (x) + rF (x) = abe , x ∈ R, −x a  dx   1 − 1 − e−e  with boundary condition limx→∞ rF (x) = 0.

Proposition 2.310. Let X : Ω → (0, 1) be a continuous random variable. The random variable X has pd f (1.1.894) if and only if its hazard function rF (x) satisfies the following differential equation rF0 (x) + αβx−1 rF (x) = −αβ (β − 1) x−2 (− logx)β−2 , 0 < x < 1,

with boundary condition limx→∞ rF (x) = 0 for β > 1.

Proposition 2.311. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.920) if and only if its hazard function rF (x) satisfies the following differential equation n 
−x/θ α−1 o  x x 1 + θ e d rF0 (x) + θ−1 rF (x) = αβθ−2 e−x/θ  
α  , x > 0, dx 1 − 1 + θx e−x/θ

with boundary condition limx→∞ rF (x) = 0 for α > 1.

Characterizations of Distributions

405

Proposition 2.312. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.926) if and only if its hazard function rF (x) satisfies the following differential equation rF0 (x) + mrF (x) = ame−mx × (   ) (1 − λ − β) + 2 (λ + 2β) e−mx − 3βe−2mx d , x > 0, dx [1 + (λ + β − 1) e−mx − (λ + 2β) e−2mx + βe−3mx ] with boundary condition limx→∞ rF (x) = 0. Proposition 2.313. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.938) if and only if its hazard function rF (x) satisfies the following differential equation

rF0 (x) −

g0 (x; η) rF (x) = abg (x; η) × g (x; η) ( ) −b d (1 − G (x; η))−b−1 e1−(1−G(x;η)) , x ∈ R, −b dx 1 − e1−(1−G(x;η))

with boundary condition limx→∞ rF (x) = 0. Proposition 2.314. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.958) if and only if its hazard function rF (x) satisfies the following differential equation

rF0 (x) −

(c − 1) g (x; η) rF (x) = ckG (x; η)c−1 × G (x; η) ( ) d g (x; η) [1 + G (x; η)c] , x ∈ R, k dx [1 + G (x; η)c ] − 1

with boundary condition limx→∞ rF (x) =

ck2−k−1 1−2−k

limx→∞ g (x; η).

Proposition 2.315. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.968) if and only if its hazard function rF (x) satisfies the following differential equation rF0 (x) −

g0 (x; η) rF (x) = −2λg (x; η)2 G (x; η)−3 , x ∈ R, g (x; η)

with boundary condition limx→∞ rF (x) = λ (1 + γ) limx→∞ g (x; η). Proposition 2.316. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.992) if and only if its hazard function rF (x) satisfies the following differential equation

G. G. Hamedani

406

rF0 (x) +



1 β

 − 1 βλ

1 − βλx

rF (x) = −αλ2 (1 − βλx)

with boundary condition limx→∞ rF (x) = 0.

2



1 β −1



×

h i 1 −2 1 − (1 − βλx) β , x > 0,

Proposition 2.317. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.1016) if and only if its hazard function rF (x) satisfies the following differential equation rF0 (x) −

g0 (x; η) rF (x) = 2bδ2 g (x; η) × g (x; η)   h i−1    1 − δG (x; η) d  h i , x ∈ R, 2 dx   1 − δG (x; η) − δ2 G (x; η)  

with boundary condition limx→∞ rF (x) = 0.

Proposition 2.318. Let X : Ω → (0, 1) be a continuous random variable. The random variable X has pd f (1.1.1068) if and only if its hazard function rF (x) satisfies the following differential equation rF0 (x) + 2x−1 rF (x) = − with boundary condition limx→1 rF (x) =

θ2 (θ + x)2 x2

, 0 < x < 1,

θ2 θ+‘1 .

Proposition 2.319. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.1112) if and only if its hazard function rF (x) satisfies the following differential equation rF0 (x) + βλxβ−1 rF (x) = αβλe−λx

β

  o d n β−1 β x cotanh α 1 − e−λx , x > 0, dx

with boundary condition limx→∞ rF (x) = 0.

Proposition 2.320. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.1196) if and only if its hazard function rF (x) satisfies the following differential equation   αθ−1  β     1 + x 2 αβθ d 0 rF (x) + rF (x) = 2 h   i1−θ  , x > 0, x x dx   1− 1+ β α  x with boundary condition limx→∞ rF (x) = 0.

Characterizations of Distributions

407

Proposition 2.321. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.1230) if and only if its hazard function rF (x) satisfies the following differential equation    x−µ σ δσ   Φ − , ; −δ B x−µ 1 d σ τ τ , x ∈ R, rF0 (x) − rF (x) = e( σ )  σ dx  A (x)

with boundary condition limx→∞ rF (x) = ∞.

Proposition 2.322. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.1272) if and only if its hazard function rF (x) satisfies the following differential equation rF0 (x) −

a (b − 1) abπ [1 − G (x)a ] × a rF (x) = 1 − G (x) 2 π o d n b g (x) G (x)a−1 tan [1 − G (x)a ] , x ∈ R, dx 2

with boundary condition limx→∞ rF (x) = ∞, for b > 1.

Proposition 2.323. Let X : Ω → (0, ∞) be a continuous random variable. The random variable X has pd f (1.1.1276) if and only if its hazard function rF (x) satisfies the following differential equation  −2 3 9 3 rF0 (x) + rF (x) = − e−3x 1 − e− 2 x , x > 0, 2 2 with boundary condition limx→∞ rF (x) = 0. Proposition 2.324. Let X : Ω → R be a continuous random variable. The random variable X has pd f (1.1.1288) if and only if its hazard function rF (x) satisfies the following differential equation g0 (x; η) rF (x) = g (x; η) × g (x; η) ( ) d β + (β − α) G (x; η)α   , x ∈ R, dx G (x; η) 1 + G (x; η)α
with boundary condition limx→∞ rF (x) = β − α2 limx→∞ g (x; η). rF0 (x) −

2.4 Characterization Based on the Conditional Expectation of Certain Function of the Random Variable In this subsection we employ a single function ψ (or ψ1 ) of X and characterize the distribution of X in terms of the truncated moment of ψ (X) (or ψ1 (X)). The following propositions have already appeared in Hamedani’s previous work (2013), so we will just state them here which can be used to characterize some of the new distributions listed in Section 1.

G. G. Hamedani

408

Proposition 2.325. Let X : Ω → (e, f ) be a continuous random variable with cd f F . Let ψ (x) be a differentiable function on (e, f ) with limx→e+ ψ (x) = 1. Then for δ 6= 1 , E [ψ (X) | X ≥ x] = δψ (x) ,

x ∈ (e, f ),

if and only if 1

ψ (x) = (1 − F (x)) δ −1 ,

x ∈ (e, f )

Proposition 2.326. Let X : Ω → (e, f ) be a continuous random variable with cd f F . Let ψ1 (x) be a differentiable function on (e, f ) with limx→ f − ψ1 (x) = 1. Then for δ1 6= 1 , implies

E [ψ1 (X) | X ≤ x] = δ1 ψ1 (x) , 1

ψ1 (x) = (F (x)) δ1

−1

.

x ∈ (e, f )

x ∈ (e, f )

Remarks 2.3.  h i−β  (A) For (e, f ) = R, ψ (x) = exp − G(x;η) and δ = G(x;η)

θβ , θβ +1

Proposition 2.325 pro-

vides a characterization of the IWG distribution.  
 1 θ (B) For (e, f ) = (0, ∞), ψ1 (x) = α+1 1 − e−λx − 1+α 1 − e−λ(1+α)x and δ1 = θ+1 , α Proposition 2.326 provides a characterization of the EWE distribution.  1/θ βθ θ (C) For (e, f ) = (0, ∞) , ψ (x) = 1 + θ+β H (x; η) e−H(x;η) and δ = θ+1 , Proposition 2.325 provides a characterization of the ACLW distribution. b(b−1)

b

a , Proposition 2.325 provides a (D) For (e, f ) = (1, ∞) , ψ (x) = x a e1−x and δ = a+b characterization of the BType II distribution. h i 2 α (E) For (e, f ) = R, ψ1 (x) = eG (x; β)−α e−G(x;β) and δ1 = α+1 , Proposition 2.326 provides a characterization of the MG distribution.

(F) For (e, f ) = (0, ∞), ψ (x) = n



  h i 1/a β α (1+a)−1 a+ 1− 1−e−x

h i o1/a β α 1− 1−e−x e

      β α β α 1−e−x / 1− 1−e−x

a and δ = a+1 , Propo-

sition 2.325 provides a characterization of the OLEW distribution.  α i  α h θ (G) For (e, f ) = (0, β), ψ1 (x) = βx 1 + γ − γ βx and δ1 = θ+1 , Proposition 2.326 provides a characterization of the ETPF distribution.  


1/α x0 x0 α x0 2α α (H) For (e, f ) = (0, ∞), ψ (x) = x 1 − λ x + λ x and δ = α+1 , Proposition 2.325 provides a characterization of the CTP distribution.

Characterizations of Distributions  (I) For (e, f ) = (0, ∞) , ψ (x) = exp − 21 − π1 arctan [ν sinh(logx)] and δ = sition 2.325 provides a characterization of the LSCp distribution.

409 τ τ+1 ,

Propo-

G(x;η) θ 1/θ and δ = θ+1 , Proposition 2.325 provides a {1−log[[G(x;η)]]} characterization of the BH-G distribution.  1/σ θ(β+σ)−σxβ σ (K) For (e, f ) = (0, 1), ψ1 (x) = x θβ+(θ−1)σ , Proposition 2.326 proand δ1 = σ+1 vides a characterization of the GLL distribution. 
−λ  α , Proposition 2.326 pro(L) For (e, f ) = R, ψ1 (x) = 1 − 1 + ex/θ and δ1 = α+1 vides a characterization of the EEL distribution.

(J) For (e, f ) = R, ψ (x) =

√

√

λx

α (M) For (e, f ) = (0, ∞) , ψ (x) = e−α x−β xe and δ = α+1 , Proposition 2.325 provides a characterization of the RNMW distribution.    β  G(x;η) −λ G(x;η) G(x;η) and δ = 12 , Proposition 2.325 (N) For (e, f ) = R, ψ (x) = exp −α G(x;η) e

provides a characterization of the EW-G distribution.  1/α (O) For (e, f ) = (0, ∞) , ψ1 (x) = x[x(1+β)+(1+α+β)β] and δ1 = 12 , Proposition 2.326 1/α 1+ 1 (1+β)

(β+x)

α

provides a characterization of the GaL distribution.

(P) For (e, f ) = (0, ∞) , ψ (x) =

a exp[−bκαβ (x)]

1−(1−a)exp[−bκαβ (x)]

and δ = 12 , Proposition 2.325 pro-

vides a characterization of the MOEFW distribution.  h n  γ o i1/α2 β α α (Q) For (e, f ) = R, ψ (x) = log e − 1 − 1 − F (x; ξ) (e − 1) and δ = 12

, Proposition (2.325) provides a characterization of LKu distribution. io n h βG(x;φ) α and δ = 12 , Propo(R) For (e, f ) = R, ψ (x) = [log(1 − β)]−1 log 1 − G(x;φ) α α +G(x;φ) sition (2.325) provides a characterization of AOLLL-G distribution. " #1/λ λ (λ2 +x−λ) 1 (S) For (e, f ) = (0, ∞), ψ (x) = e−x and δ = 1+λ , Proposition 2.325 λ(λ−1)

provides a characterization of ADFIF distribution. h λ ia b (T ) For (e, f ) = R, ψ (x) = 1 − 1−G(x)λ and δ = 1+b , Proposition 2.325 provides a 1+G(x) characterization of KwTIHL-G distribution.  1/θ θx(θ2 x2 +3θx+6) θ (U) For (e, f ) = (0, ∞), α = 1, ψ (x) = 1 + e−x and δ = 1+θ , Propoθ4 +6 sition 2.325 provides a characterization of EP distribution.   θx(θ2 x2 +3θx+6) (V ) For (e, f ) = (0, ∞) , ψ1 (x) = 1 − 1 + e−θx and δ1 = θ4 +6 tion 2.326 provides a characterization of EP distribution.

α 1+α

, Proposi-

G. G. Hamedani

410

 (W ) For (e, f ) = R, ψ (x) = 1 +

λG(x;ϕ)α (1+λ)[1−G(x;ϕ)α ]

1/λ

exp

h

−G(x;ϕ)α 1−G(x;ϕ)α

i

and δ =

λ 1+λ

,

Proposition 2.325 provides a characterization of GOLi-G distribution. n o γ τ (X) For (e, f ) = (θ, ∞), ψ (x) = exp − (x−θ) and δ = 1+τ , Proposition 2.325 provides x a characterization of LDSWeibull distribution. (Y ) For (e, f ) = (0, 1), ψ1 (x) =

x(1−m)

[xb (1−m)b +mb (1−x)b ]

1/b

and δ1 =

b 1+b

, Proposition 2.326

provides a characterization of L-Logistic distribution. "

2

(Z) For (e, f ) = (0, 1), ψ (x) = 1 +

2 λ2 +λ x

(

x λ2 ( 1−x ) + 1−x λ2 +2λ+2

)

#1/λ

x

e− 1−x and δ =

λ 1+λ

, Proposi-

tion 2.325 provides a characterization of unit-ISDL distribution. 0

"

−xβ

(A ) For (e, f ) = (0, ∞), ψ (x) = e

0

(B ) 0

(C ) 0

(D )

0

(E )

0

(F )

0

θβ /δ1 #δ1 /θβ  x β and δ = 1 − λ + λe−δ1 ( θ )

δ1 δ1 +θβ

,

Proposition 2.325 provides a characterization of TNWP distribution.   b For (e, f ) = R, ψ1 (x) = 1−21 −a 1 − (1 + G (x; η))−a and δ1 = 1+b , Proposition 2.326 provides a characterization of TIK-G distribution. io−1 n h  −θx− αβ (eβx−1) a and δ = 1+a For (e, f ) = (0, ∞), ψ (x) = b b − log 1 − 1 − e , Proposition 2.325 provides a characterization of LOGOMA distribution. h  i α α θ For (e, f ) = (0, ∞) , ψ (x) = 1 − e−(β/x) 1 + λ 1 − e−(β/x) and δ = 1+θ , Proposition 2.325 provides a characterization of ETF distribution.   −α 1/β −α β For (e, f ) = (0, ∞), ψ (x) = β+λ+βλx e−x and δ = 1+β , Proposition 2.325 β+λ provides a characterization of GIL distribution. h α n
α io1/θ −(− log G(x;φ)) θ For (e, f ) = R, ψ (x) = e 1 + θ 1 + − log G (x; φ) and δ = 1+θ 1/θ (1+θ) , Proposition 2.325 provides a characterization of EPL-G distribution.   a α − x − − βc ln 1−e ( b )

θ (G ) For (e, f ) = (0, ∞) , ψ (x) = 1−e and δ = 1+θ , Proposition 2.325 provides a characterization of EW-EW distribution.  i h 0 β a λxβ a (H ) For (e, f ) = (0, ∞) , ψ (x) = 1 + 1+λ e−λx and δ = 1+a , Proposition 2.325 provides a characterization of EG-PL (for b = 1) distribution. n
−1/σo 0 α (I ) For (e, f ) = (0, ∞) , ψ (x) = 1 − exp −B eθx − 1 and δ = 1+α , Proposition 2.325 provides a characterization of EGuE distribution. n
−1/σo 0 B (J ) For (e, f ) = (0, ∞), α = 1, ψ1 (x) = exp − eθx − 1 and δ1 = 1+B , Proposition 2.326 provides a characterization of EGuE (for α = 1) distribution.

Characterizations of Distributions 

0

1−

h

 1+ βx e−x/β

411

i−γ 

θ and δ = θ+γ , Proposition 2.325 (K ) For (e, f ) = (0, ∞) , ψ (x) = e provides a characterization of Go-LBE distribution. n o1/β 0 αβ (L ) For (e, f ) = R, ψ (x) = [1 − G (x; η)] 2 − [1 − G (x; η)]β and δ = 1+αβ , Proposition 2.325 provides a characterization of TIIPTL-G distribution. 0

−

n

oθ

G(x;ζ)

λ (M ) For (e, f ) = R, ψ (x) = e 1−G(x;ζ)[1+G(x;ζ)]/2 and δ = 1+λ , Proposition 2.325 provides a characterization of MOW-G distribution.  −1 0 (1+a)eax β (N ) For (e, f ) = (0, ∞) , ψ (x) = α(1+a+ax) − αα and δ = 1+β , Proposition 2.325 provides a characterization of WMOL distribution. h
−α
α i 0 β (O ) For (e, f ) = (a, b), ψ (x) = 1 − ln ab ln ax and δ = 1+β , Proposition 2.325 provides a characterization of KR distribution. 1/α  −α  0 1+α logx α (P ) For (e, f ) = (β, ∞), ψ (x) = βx and δ = 1+α , Proposition 2.325 1+α logβ provides a characterization of LWP distribution. G(x)

λG(x)

1/λρ

λρ (Q ) For (e, f ) = R, ψ (x) = e e(e(eλ −−1)−1) and δ = 1+λρ , Proposition 2.325 provides a 1/λρ characterization of IPC distribution. (  β ) −λ/x 0 ϕ αe −1 (R ) For (e, f ) = (0, ∞), ψ (x) = exp − and δ = 1+ϕ , Proposition 2.325 e−λ/x 0

α−α

provides a characterization of WAPIE distribution.
0 λd d+x −λd/2 (S ) For (e, f ) = (0, d), ψ1 (x) = 1 − d−x , Proposition 2.326 proand δ1 = 2+λd vides a characterization of E-Epsilon distribution.
0 d+x −1 α (T ) For (e, f ) = (0, d) , α = 1, ψ (x) = d−x and δ = 1+α , Proposition 2.325 provides a characterization of E-Epsilon distribution.   b  0 α−1 a (U ) For (e, f ) = (0, ∞) , ψ (x) = exp 1 − , Proposition 2.325 and δ = a+b e−c/x α−α

provides a characterization of GAPIE distribution. h γ iλ 0 β −(β−1)γ (V ) For (e, f ) = (0, ∞) , ψ (x) = 1 − exp βγ −(β−e−x )γ and δ = provides a characterization of G-EGE distribution.

(W ) For (e, f ) = (0, ∞) , ψ (x) = γ[a,ρ(x)] and δ = Γ(a) acterization of RBEE distribution. 0

0

−x

1 2

θ θ+λ

, Proposition 2.325

, Proposition 2.325 provides a char-

a (X ) For (e, f ) = R, b = 1, ψ (x) = 1 − e−e and δ = 1+a , Proposition 2.325 provides a characterization of EGSGu distribution.  1/θ x 0 θx θ (Y ) For (e, f ) = R, ψ (x) = 1 − (1+θ)(x−1) e− 1−x and δ = 1+θ , Proposition 2.325 provides a characterization of unit-Lindley distribution.

G. G. Hamedani h i 0 α , Proposition 2.325 pro(Z ) For (e, f ) = (0, 1), ψ (x) = exp − (− log x)β and δ = 1+α vides a characterization of UW distribution.
00 α , Proposition 2.325 provides (A ) For (e, f ) = (0, ∞), ψ (x) = 1 + θx e−x/θ and δ = 1+α a characterization of ELBE distribution.   00 (B ) For (e, f ) = (0, ∞) , ψ1 (x) = 1 + (λ + β − 1) e−mx − (λ + 2β) e−2mx + βe−3mx and a δ1 = 1+a , Proposition 2.326 provides a characterization of ELBE distribution.

412

00

−b

a , Proposition 2.326 provides (C ) For (e, f ) = R, ψ1 (x) = 1 − e1−(1−G(x;η)) and δ1 = 1+a a characterization of ETIW-G distribution. 00

(D ) For (e, f ) = R, ψ1 (x) = exp {λ [1 − G (x; η)][1 + γG (x; η)] / G (x; η)} and δ1 = , Proposition 2.326 provides a characterization of GOIE-G distribution.

λ 1+λ

00

1 , Proposition 2.325 (E ) For (e, f ) = (0, ∞), α = 1, ψ1 (x) = (1 − βλx) and δ = 1+β provides a characterization of EEW distribution.   00 (F ) For (e, f ) = R, b = 1, ψ (x) = δG(x;η) and δ = 32 , Proposition 2.325 provides a 1−δG(x;η) characterization of TLMO-G distribution.   1/2  1/2  00 log x β 1 (G ) For (e, f ) = (0, ∞), ψ1 (x) = Φ α − β − − log x and δ1 = 12 ,

Proposition 2.326 provides a characterization of UBS distribution.  1/θ

00 (θ+x) θ (H ) For (e, f ) = (0, 1), ψ1 (x) = x(1+θ) exp − 1−x and δ1 = 1+θ , Proposition x 2.326 provides a characterization of NUL distribution. h  i2 α α +2) 00 α β (I ) For (e, f ) = (0, ∞) , ψ (x) = 1 − 1 − 1 + θx θ(θx e−2θx and δ = 1+β , Propo3 +2 sition 2.325 provides a characterization of TIITLPI-G distribution.   2   00 aG(x;φ)c G(x;φ)c b and δ = 12 , Proposi(J ) For (e, f ) = R, ψ (x) = exp − 1−G(x;φ)c + 2 1−G(x;φ)c

tion 2.325 provides a characterization of GOLE distribution.   β β −1 00 k (K ) For (e, f ) = (0, ∞) , ψ (x) = (1 − p) e−(xα) 1 − pe−(xα) and δ = 1+k , Proposition 2.325 provides a characterization of WNB distribution. !−1 −1/λ −θ+θ(1−λxθ ) 00 β −1 (L ) For (e, f ) = (0, ∞), ψ (x) = 1 + e and δ = 1+β , Proposition eθ −1

2.325 provides a characterization of LDGW distribution.   00 − 23 x and δ1 = 23 , Proposition 2.326 provides a (M ) For (e, f ) = (0, ∞), ψ1 (x) = 1 − e characterization of WE distribution. 
−1  00 −a 1/α α (N ) For (e, f ) = (0, ∞) , ψ (x) = 1 + xβ 1 − e−bx and δ = 1+α , Proposition 2.325 provides a characterization of minGuBu distribution.

Characterizations of Distributions i h β 00 (O ) For (e, f ) = (0, ∞), ψ (x) = exp − 1k eαx− x [1 + xc ]−1 and δ = 2.325 provides a characterization of FWBXII distribution.

413 k 1+k

, Proposition

Appendix Theorem 2.1. Let (Ω, F , P) be a given probability space and let H = [a, b] be an interval for some d < b (a = −∞, b = ∞ might as well be allowed). Let X : Ω → H be a continuous random variable with the distribution function F and let q1 and q2 be two real functions defined on H such that

E [q2 (X) | X ≥ x] = E [q1 (X) | X ≥ x] ξ (x) ,

x ∈ H,

is defined with some real function η. Assume that q1 , q2 ∈ C1 (H), ξ ∈ C2 (H) and F is twice continuously differentiable and strictly monotone function on the set H. Finally, assume that the equation ξq1 = q2 has no real solution in the interior of H. Then F is uniquely determined by the functions q1 , q2 and ξ , particularly

F (x) =

Z x a

ξ0 (u) exp (−s (u)) du , C ξ (u)q1 (u) − q2 (u)

0

q1 where the function s is a solution of the differential equation s0 = ξqξ1 −q and C is the 2 R normalization constant, such that H dF = 1. Note: The goal is to have the function ξ(x) as simple as possible. We like to mention that this kind of characterization based on the ratio of truncated moments is stable in the sense of weak convergence (see Glänzel, 1990), in particular, let us assume that there is a sequence {Xn } of random variables with distribution functions {Fn } such that the functions q1n , q2n and ξn (n ∈ N) satisfy the conditions of Theorem 2.1 and let q1n → q1 , q2n → q2 for some continuously differentiable real functions q1 and q2 . Let, finally, X be a random variable with distribution F . Under the condition that q1n (X) and q2n (X) are uniformly integrable and the family {Fn } is relatively compact, the sequence Xn converges to X in distribution if and only if ξn converges to ξ , where

ξ (x) =

E [q2 (X) | X ≥ x] . E [q1 (X) | X ≥ x]

This stability theorem makes sure that the convergence of distribution functions is reflected by corresponding convergence of the functions q1 , q2 and ξ , respectively. It

guarantees, for instance, the ‘convergence’ of characterization of the Wald distribution to that of the Lévy-Smirnov distribution if α → ∞. A further consequence of the stability property of Theorem 2.1 is the application of this theorem to special tasks in statistical practice such as the estimation of the parameters of discrete distributions. For such purpose, the functions q1 , q2 and, specially, ξ should be as simple as possible. Since the function triplet is not uniquely determined it is often possible to choose ξ as a linear function. Therefore, it is worth analyzing some special cases which helps to find new characterizations reflecting the relationship between individual continuous univariate distributions and appropriate in other areas of statistics. In some cases, one can take q1 (x) ≡ 1, which reduces the condition of Theorem 2.1 to E [q2 (X) | X ≥ x] = ξ (x) , x ∈ H. We, however, believe that employing three functions q1 , q2 and ξ will enhance the domain of applicability of Theorem 2.1.

References Abayomia, A. and M. Adelele (2019). Transmuted half normal distribution: properties and application. Mathematical Theory and Modeling 9(1), 13–26. Abbas, S., M. Hameed, S. Cakmakyapan, and S. N. Malik (2019). On gamma inverse weibull distribution. Journal Natn. Sci 47(4), 445–453. Abbas, S., M. Mohsin, and J. Pilz (2020). A new life time distribution with applications in reliability and environmental sciences. Journal of Statistics and Management Systems, 1–27. Abbas, S., G. Ozel, S. H. Shahbaz, and M. Shahbaz (2019). A new generalized weighted weibull distribution. Pak. Journal stat. oper. res 15(1), 161–178. Abbas, S., S. A. Taqi, F. Mustafa, M. Murteza, and M. Shahbaz (2017). Topp-leone inverse weibull distribution: theory and applications. European Journal of Pure and Applied Mathematics 10(5), 1005–1022. Abd AL-Kadim, K. and M. Mohammed (2017). The cubic transmuted weibull distribution. Journal of Babylon University/Pure and Applied Sciences 25(3), 862–876. Abd El-Bar, A. and I. Ragab (2020). On weighted exponential-gompertz distribution: properties and application. Journal of Taibah University for Science (forthcoming). Abd Elrazik, E. and M. Mansour (2019). A new generated distribution to analyze a practical engineering problem and application. Journal of Nonlinear Sci. Appl 12(7), 470–484. Abdelall, Y. Y. (2017). The beta transmuted weighted exponential distribution. International Journal of Mathematical Archive 8(11), 16–28. Abdelall, Y. Y. (2019a). The modified weibull-g family of distributions: properties and applications. International Journal of Mathematical Archive 10(4), 21–31. Abdelall, Y. Y. (2019b). A new generalization of lindley distribution. JDS 17(3), 631–642. Abdelkhalek, R. H. M. (2018). Extended poisson-burr XII distribution. Journal of Statistics and Applications 1(2), 46–70. Abdelkhalek, R. H. M. (2019). Extended poisson-pareto type II distribution: theoretical and computational aspects. PJSOR XV(3), 713–730.

418

References

Abdelkhalek, R. H. M. (2020). The poisson burr x inverse rayleigh distribution and its applications. JDS 18(1), 56–77. Abdi, M., M. Afshari, H. Karamikabir, M. Mozafari, and M. Alizadeh (2019). The new odd log-logistic generalized half-normal distribution: mathematical properties and simulations. PJSOR 15(2), 277–302. Abdollahi, A., S. M. T. K. MirMostafaee, and E. Altun (2019). A new two-parameter distribution: properties and applications. Journal of Mathematical Modeling (JMM) 7(1), 35–48. Abdul-Moniem, I. B. and L. S. Diab (2018). Generalized transmuted power function distribution. Journal Stat. Appl. Pro 7(3), 401–411. Abdul-Moniem, I. B. and M. Seham (2019). Type II logistic ibrahim distribution with applications. International Journal of Advanced Statistics and Probability (JASP) 7(1), 1–6. Abdulkabir, M. and R. A. Ipinyomi (2020). Type II half logistic exponentiated exponential distribution: properties and applications. Pak. Journal Statist. 36(1), 29–56. Abid, S. H., N. H. Al-Noor, and A. A. M. Boshi (2018). The generalized gammageneralized inverse weibull distribution. Journal of Iraqi Al-Khwarizmi Society (JIKhS) 2, 149–158. Abid, S. H., N. H. Al-Noor, and A. A. M. Boshi (2019). On the generalized inverse weibull distribution. International Conference of Mathematical Sciences (ICMS 2018). Abid, S. H., N. H. Al-Noor, and A. A. M. Boshi (2020). The generalized gammaexponentiated weibull distribution with its properties. Al-Mustansiriyah Journal of Science (AJS) 31(2), 30–37. Aboray, M. and N. S. Butt (2019). Extended weibull burr XII distribution: properties and applications. PJSOR XV(4), 891–903. Aboraya, M. (2018). A new flexible lifetime model with statistical properties and applications. PJSOR XIV(4), 881–901. Aboraya, M. (2019). A new extremely flexible version of the exponentiated weibull model: theorem and applications to reliability and medical data sets. PJSOR XV(1), 195–215. Abouelmagd, T. H. M. (2019). A new flexible distribution based on the zero truncated poisson distribution: mathematical properties and applications to lifetime data. Biostatistics and Biometrics Open Access Journal (BBOAS) 8(1), 1–7. Abouelmagd, T. H. M., M. S. Hamed, J. A. Almamy, M. M. Ali, H. M. Yousof, and M. C. Korkmaz (2019). Extended weibull log-logistic distribution. Journal Nonlinear Sci. Appl 12, 523–534. Abouelmagd, T. H. M., M. S. Hamed, and H. M. Yousof (2019). Poisson burr x weibull distribution. Journal of Nonlinear Science and Applications 12, 173–183.

References

419

Abu-Youssof, S. E., B. I. Mohammed, and M. G. Sief (2015). An extended exponentiated exponential distribution and its properties. International Journal of Computer Applications 121(5), 1–6. Acitas, S., T. Arslan, and B. Senoglu (2020). Slash maxwell distribution: definition, modified maximum likelihood estimation and applications. GU J Sci 33(1), 249–263. Adeleke, M., A. A. Ayodele, and A. F. Barnabas (2019). Exponentiated exponential lomax distribution and its properties. Mathematical Theory and Modeling 9(1), 1–13. Ademola, A. J. and A. J. Sunday (2020). The comparative study of gompertz exponential distribution and other three parameter distributions of exponential class. Covenant Journal of Physical & Life Sciences (CJPL) 8(1), 1–13. Adeyinka, F. S. (2019a). On the tractability of transmuted type I generalized logistic distribution with application. International Journal of Theoretical and Applied Mathematics 5(2), 31–36. Adeyinka, F. S. (2019b). On transmuted type II generalized logistic distribution with application. American Journal of Applied Mathematics 7(6), 190–195. Adeyinka, F. S. (2020). On modelling of infant mortality rate in nigeria with exponentiated cubic transmuted exponential distribution. International Journal on Data Science and Technology 6(1), 16–22. Afify, A. Z. (2020). Three-parameter exponential distribution: estimation and applications. (Forthcoming). Afify, A. Z. and A. D. Abdellatif (2020). The extended burr XII distribution: properties and applications. Journal of Nonlinear Sci. Apll. 13, 133–146. Afify, A. Z. and O. Abdo Mohamed (2020). A new three-parameter exponential distribution with variable shapes for hazard rate: estimation and applications. Mathematics 8, 135, 1–17. Afify, A. Z., H. Al-Mofleh, and S. Dey (2019). Topp-leone odd log-logistic exponential distribution: its improved estimators and applications. Anais da Academia Brasileira de Ciencias. (Forthcoming). Afify, A. Z., M. Alizadeh, M. Zayed, T. G. Ramires, and F. Louzada (2018). The odd log-logistic exponentiated weibull distribution: regression modeling, properties and applications. Iran Journal Sci. Technol. Trans. Sci 42(4), 2273–2288. Afify, A. Z., E. Altun, M. Alizadeh, G. Ozel, and G. G. Hamedani (2017). The odd exponentiated half-logistic-g family: properties, characterizations and applications. Chilean Journal of Statistics 8, 65–91. Afify, A. Z., G. M. Cordeiro, F. Jamal, M. Elgarhy, and M. Nasir (2019). The marshallolkin odd burr IIi-g family of distributions: theory, estimation and applications. HAL Id: hal-02376067, 1-22.

420

References

Afify, A. Z., G. M. Cordeiro, M. E. Maed, M. Alizadeh, F. Al-Mofleh, and Z. M. Nofal (2019). The generalized odd lindley-g family: properties and applications. Annals of the Brazilian Academy of Sciences 91(3), 1–22. Afify, A. Z., G. M. Cordeiro, S. Nadarajah, H. M. Yousof, G. Ozel, M. Nofal, and E. Altun (2017). The complementary geometric transmuted-g family of distributions: model, properties and application. Hacettepe Journal of Mathematics and Statistics 47(5), 1348– 1374. Afify, A. Z., A. M. Gemeay, and N. A. Ibrahim (2020). The heavy-tailed exponential distribution: risk measures, estimation, and application to actuarial data. Mathematics (MDPI) 8(8), 1–28. Afify, A. Z., D. Kumar, and I. Elbatal (2020). Marshall-olkin power generalized weibull distribution with applications in engineering and medicine. JSTA 19(2), 223–237. Afify, A. Z., M. Nassar, G. M. Cordeiro, and D. Kumar (2020). The weibull marshallolkin lindley distribution: properties and estimation. Journal of Taibah University for Science 14(1), 192–204. Afify, A. Z., A. K. Suzuki, C. Zhang, and M. Nassar (2020). On three-parameter exponential distribution: properties, Bayesian and non-bayesian estimation based on complete and censored samples. Communications in Statistics-Simulation and Computation, 1–21. Afify, A. Z., M. Zayed, and M. Ahsanullah (2018). The extended exponential distribution and its applications. JSTA 17(2), 213–229. Afify, A, Z. and M. Alizadeh (2020). The odd dagum family of distributions: properties and applications. Journal of Applied Probability and Statistics 15(1), 45–72. Aguilar, G. A. S., F. A. Moala, and G. M. Cordeiro (2019). Zero-truncated poisson exponentiated gamma distribution: application and estimation methods. Journal of Statistical Theory and Practice (JSTP) 13(4), 57. Ahmad, A., P. B. Ahmad, and I. S. Ahmad (2019). Weighted analogue of inverse lévy distribution: statistical properties and estimation. Journal of Applied Probability and Statistics (JAPS) 14(2), 99–112. Ahmad, H. A. H. and E. M. Almetwally (2020). Marshall-olkin generalized pareto distribution: Bayesian and non Bayesian estimation. PJSOR 16(1), 21–33. Ahmad, Z., R. A. and T. R. Jan (2018). Complementary compound lindley power series distribution with application. PJSOR XIV(1), 139–155. Ahmad, Z. (2019a). The hyperbolic sine rayleigh distribution with application to bladder cancer susceptibility. Annals of Data Science 6(2), 211–222. Ahmad, Z. (2019b). The logarithmic kumarswamy family of distributions: properties and applications. Communications of the Korean Mathematical Society 34(4), 1335–1352.

References

421

Ahmad, Z. (2020a). The new generalized class of distributions: properties and estimation based on type-i censored samples. AODS 7(2), 243–256. Ahmad, Z. (2020b). The zubair-g family of distributions: properties and applications. AODS 7(2), 195–208. Ahmad, Z., M. Elgarhy, and N. Abbas (2018). A new extended alpha power transformed family of distributions: properties and applications. Journal of Statistical Modelling: Theory and Applications 1(2), 13–28. Ahmad, Z., G. G. Hamedani, and P. Algamgir (2018). Type-i alpha power transformed family of distributions: properties, characterizations and applications. (Submitted). Ahmad, Z., S. M. Hamid, and Z. Hussain (2017). Generalized new extended weibull distribution with real life application. MASYFEB Journal of Environmental Science 3, 1–11. Ahmad, Z., E. Mahmoudi, and M. Alizadeh (2020a). Modelling insurance losses using a new beta power transformed family of distributions. Communications in StatisticsSimulation and Computation, 1–22. Ahmad, Z., E. Mahmoudi, and M. Alizadeh (2020b). On modelling the automobile insurance claims via a new heavy tailed family of claim distributions. (Personal Communication). Ahmad, Z., E. Mahmoudi, and Z. Almaspoor (2019). A new family of type-i heavy tailed distributions: properties, regression theory with real life applications to different disciplines. (Forthcoming). Ahmad, Z., E. Mahmoudi, S. Dey, and S. Khosa (2020). Modeling vehicle insurance loss data using a new member of t-x family of distributions. JSTA 19(2), 133–147. Ahmed, M. A. (2020). The new form of libby-novick distribution. Communications in Statistics-Theory and Methods. Ahsan ul Haq, M., A. A. H. A. and M. Elgarhy (2018a). Type II half logistics rayleigh distribution: properties, estimation based on censored sample. Advances in Mathematics and Computer Science (JAMCS) 29(1), 1–19. Ahsan ul Haq, M., H. S. A. K. R. P. and F. Louzada (2020). Unit modified burr-III distribution: estimation, characterizations and validation test. Annals of Data Science, 1–26. Ahsan ul Haq, M., A. Afify, H. Al-Mofleh, and R. Usman (2019, Journal of modern applied statistical methods). The marshall-olkin modified burr III distribution: properties and applications. Ahsan ul Haq, M., N. Butt, R. Usman, and A. Fattah (2016). Transmuted power function distribution. Gazi University Journal of Science 29(1), 177–185. Ahsan ul Haq, M. and M. Elgarhy (2018b). The odd fréchet-g family of probability distributions. Journal of Statistics Applications & Probability 7(1), 189–203.

422

References

Ahsan ul Haq, M., M. Elgarhy, and S. Hashmi (2019). The generalized odd burr III family of distributions: properties, applications and characterizations. Journal of Taibah University for Science 13(1), 961–971. Ahsan ul Haq, M., W. Marzouk, S. Hashmi, and T. de Andrade. The generalized odd fréchet family of distributions: properties, characterizations and applications. (Personal Communication). Ahsan ul Haq, M., U. R. H. S. and A. Al-Omari (2019). The marshall-olkin lengthbiased exponential distribution and its applications. Journal of King Saud UniversityScience 31(2), 246–251. Ahsanullah, M. and V. B. Nevzorov (2019). Some characterizations of skew t-distribution of three degrees of freedom. Calcutta Statistical Association Bulletin 71(1), 40–48. Ahsanullah, M. and M. Shakil (2018). Some characterizations of raised cosine distribution. International Journal of Advanced Statistics and Probability 6(2), 42–49. Ahsanullah, M., M. Shakil, and B. M. G. Kibria (2019). On a generalized raised cosine distribution: some properties, characterizations and applications. Moroccan Journal of Pure and Appl. Anal. (MJPAA) 5(1), 63–85. Aijaz, A., M. Jallal, S. Ul Ain, and R. Tripathi (2020). The hamza distribution with statistical properties and applications. AJPAS 8(1), 28–42. Al Abbasi, J. N., M. A. Khaleel, L. Y. F. Abdal-hammed, M. Kh., and G. Ozel (2019). A new uniform distribution with bathtub-shaped failure rate with simulation and application. Mathematical Sciences 13, 105–114. Al-Babtain, A. A. (2020). A new extended rayleigh distribution. Journal of King Saud University - Science 32(5), 2576–2581. Al-Babtain, A. A., A. A. A. Fattah, N. A-hadi, and F. Merovci (2018). The kumaraswamytransmuted exponentiated modified weibull distribution. Communications in StatisticsSimulation and Computation 46(5), 3812–3832. Al-Khazaleh, A. M. H. (2016). Transmuted burr type XII distribution: A generalization of the burr type XII distribution. International Mathematical Forum 11(12), 547–556. Al-Marzouki, S. (2019). Truncated weibull power lomax distribution: statistical properties and applications. Journal of Nonlinear Sci. Appl 12(8), 543–551. Al-Marzouki, S., F. Jamal, C. Chesneau, and M. Elgarhy (2020). Type II topp leone power lomax distribution with applications. Mathematics (MDPI) 8(4), 1–26. Al-Mofleh, H. (2019). The normal-generalized hyperbolic secant distribution: properties and applications. (Personal Communication). Al-mualim, S. (2019). Extended poisson inverse weibull distribution: theoretical and computational aspects. IJSP 8(2), 146–158.

References

423

Al-Noor, N. H., S. H. Abid, and M. Abd Alhussein (2019). On the exponentiated weibull distribution. AIP Conf. Proc. 2183; DOI: 10.1063/1.5136220. Al-Omari, A. I., A. M. H. Al-khazaleh, and L. M. Alzoubi (2019). A generalization of the new weibull-pareto distribution. Revist Investigacion Operacional. 41(1), 138–146. (accepted for publication). Al-Omari, A. I. and I. K. Alsmairan (2019). Length-biases suja distribution and its applications. Journal of Applied Probability and Statistics (JAPS) 14(3), 95–116. Al-Omari, A. I. and M. M. Gharaibeh (2018). Topp-leone mukherjee-islam distribution: properties and applications. Pak. Journal Statist. 34(6), 479–494. Al-Salafi, A. A. and S. A. Adham (2018). Maximum likelihood estimation in the odd generalized exponential-exponential distribution. International Journal of Contemporary Mathematical Sciences 13(3), 111–123. Al-Zahrani, B., A. A. Fattah, S. Nadarajah, and A.-H. N. Ahmad (2015). The exponentiated transmuted weibull geometric distribution with application in survival analysis. Communications in Statistics-Simulation and Computation 46(6), 4244–4263. Aldahlan, M. A. (2020). Alpha power transformed log-logistic distribution with application to breaking stress data. Advances in Mathematical Physics 2020, 9. Aldahlan, M. A. D. and A. Z. Afify (2020). A new three-parameter exponential distribution with applications in reliability and engineering. Journal Nonlinear Sci. Appl 13, 258– 269. Aldahlan, M. A. D., A. Z. Afify, and A. H. N. Ahmed (2019). The odd inverse pareto-g class: properties and applications. Journal of Nonlinear Sci. Appl 12(5), 278–290. Aldahlan, M. A. D., F. Jamal, C. Chesneau, I. Elbatal, and M. Elgarhy (2020). Exponentiated power generalized weibull power series family of distributions: properties, estimation and applications. PLOS ONE 15(3). Aldahlan, M. A. D., F. Jamal, C. Chesneau, M. Elgarhy, and I. Elbatal (2020). The truncated cauchy power family of distributions with inference and applications. Entropy (MDPI) 22, 346, 1–24. Aldahlan, M. A. D., M. G. Khalil, and A. Z. Afify (2018). A new generalized family of distributions for lifetime data. Journal of Modern Applied Statistical Methods (JMASM). (Forthcoming). Alexopoulos, A. (2019). One-parameter weibull-type distribution, its relative entropy with respect to weibull and a fractional two-parameter exponential distribution. STATS 2, 34–54. Alghamedi, A., S. Dey, D. Kumar, and S. A. Dobbah (2020). A new extension of extended exponential distribution with applications. AODS 7(1), 139–162.

424

References

Ali, M. M., M. C. Korkmaz, H. M. Yousof, and N. S. Butt (2019). Odd lindley-lomax model: statistical properties and applications. PJSOR 15(2), 419–430. Ali, M. M., H. M. Yousof, and M. Ibrahim (2020a). Expanding the burr x model: properties, copula, real data modeling and different methods of estimation. (Personal Communication). Ali, M. M., H. M. Yousof, and M. Ibrahim (2020b). A new version of the generalized rayleigh distribution with copula properties. applications and different methods of estimation. (Personal Communication). Ali, S., S. Dey, M. H. Tahir, and M. Mansoor (2020). Two-parameter logistic-exponential distribution: some new properties and estimation methods. American Journal of Mathematical and Management Sciences 29, 1–29. Ali, Z., A. Ali, and G. Ozel (2020). A modification in generalized class of distributions: a new topp-leone class as an example. Commun. Statist. Theo-Meth. (Forthcoming). Alizadeh, M., A. Z. Afify, M. S. Eliwa, and A. Ali (2020). The odd log-logistic lindley-g family of distributions: properties, Bayesian and non-bayesian estimation with applications. Computational Statistics 35(1), 281–308. Alizadeh, M., M. Afshari, B. Hosseini, and T. G. Ramires (2020). Extended exp-g family of distributions: properties, applications and simulation. Communications in StatisticsSimulation and Computation 49(7), 1730–1745. Alizadeh, M., E. Altun, A. Z. Afify, and G. Ozel (2019). The extended odd weibull-g family: properties and applications. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68(1), 161–186. Alizadeh, M., S. F. Bagheri, B. Samani, S. Ghobagi, and S. Nadarajah (2019). Exponentiated power lindley power series class of distributions: theory and applications. 47(9), 2499–2531. Alizadeh, M., L. Benkhalifa, M. Rasekhi, and B. Hosseini (2019). The odd log-logistic generalized gompertz distribution: properties, applications and different methods of es˘ S317. timation. Communications in mathematics and statistics 8, 295âA ¸ Alizadeh, M., G. M. Cordeiro, A. D. C. Nascimento, M. D. C. Lima, and E. M. M. Ortega (2017). Odd-burr generalized family if distributions with some applications. Journal of Statistical Computation and Simulation 87(2), 367–389. Alizadeh, M., G. M. Cordeiro, L. G. B. Pinho, and I. Ghosh (2017). The gompertz-g family of distributions. Journal Statistical Theory and Practice 11, 179–207. Alizadeh, M., M. Emadi, and M. Doostparast (2019). A new two-parameter lifetime distribution: properties, applications and different method of estimation. Statistics, Optimization and Information Computing 7, 291–310.

References

425

Alizadeh, M., M. Emadi, M. Doostparast, M. H. Tahir, M. Mansoor, and G. M. Cordeiro (2016). The odd fréchet-g family of distributions. (persona communication). Alizadeh, M., M. C. Korkmaz, J. Almamy, and A. A. Ahmed (2018). Another odd loglogistic logarithmic class of continuous distributions. Journal of Statistics and Actuarial Sciences 2, 55–72. Alizadeh, M., F. Lak, M. Rasekhi, T. G. Ramires, H. M. Yousof, and E. Altun (2017). The odd log-logistic topp-leone g family of distributions: heteroscedastic regression models and applications. Computational Statistics 33(3), 1217–1244. Alizadeh, M., A. MirMostafaee, S.M.T.K., G. E., Ozel, and M. Khan Ahmadi (2018). The odd log-logistic marshall-olkin power lindley distribution: properties and applications. Journal of Statistics and Management Systems 20(6), 1065–1093. Alizadeh, M., T. G. Ramires, S. M. T. K. MirMostafaee, M. Samizadeh, and E. M. M. Ortega (2018). A new useful four-parameter extension of the gumbel distribution: properties, regression model and applications using the gamlss framework. Commun. Statist. Simul. and Comp 48(6), 1746–1767. Alizadeh, M., S. Tahmasebi, and H. Haghbin (2018). The exponentiated odd log-logistic family of distributions: properties and applications. Journal of Statistical Modelling: Theory and Applications 1(2), 29–54. Alizadeh, M., R. M. Yousof, H. M., and E. Altun. The odd log-logistic poisson-g family of distributions. Journal of Mathematical Extension 12(3), 81–104. Aljarrah, M. A., F. Famoye, and C. Lee (2019). A new generalized normal distribution: properties and applications. Commun. Statist. Theo-Meth. 48(18), 4474–4491. Aljouiee, A., I. Elbatal, and H. Al-Mofleh (2018). A new five-parameter lifetime model: theory and applications. PJSOR 14(2), 403–420. Alkarni, S. H. (2016). A class of lindley and weibull distributions. Open Journal of Statistics 6, 685–700. Alkarni, S. H. (2019). Generalized inverse lindley power series distributions: modeling and simulation. Journal of Nonlinear Sci. Appl 12, 799–815. Alkarni, S. H. and W. A. Alshehri (2020). Power lindley geometric distribution: a new model for failure analysis in business. Advances and Applications in Statistics 61(1), 1–18. Almalki, S. J. (2018). A reduced new modified weibull distribution. Communications in Statistics-Theory and Methods 47(10), 2297–2313. Almamy, J. A. (2019). Extended poisson-log-logistic distribution. IJSP 8(2), 56–69. Almarashi, A. M. and M. Elgarhy (2018). A new muth generated family of distributions with applications. Journal of Nonlinear Sciences and Applications 11, 1171–1184.

426

References

Almarashi, A. M., M. Elgarhy, M. M. Elsehetry, B. M. Kibria, and A. Algarni (2019). A new extension of exponential distribution with statistical properties and applications. Journal Nonlinear Sci. Apll 12, 135–145. Alrajhi, S. (2019). The odd fréchet inverse exponential distribution with application. Journal of Nonlinear Sci. Appl 12(8), 535–542. Altun, E. (2019a). The generalized gudermannian distribution: inference and volatility modelling. Statistics 53(2), 364–386. Altun, E. (2019b). Weighted-exponential regression model: an alternative to the gamma regression model. International Journal of Modeling Simulation and Scientific Computing 10(6), 15 pages. Altun, E. (2020). The log-weighted exponential regression model: alternative to the beta regression model. Communications in Statistics-Theory and Methods, 1–16. Altun, E., M. Alizadeh, G. Ozel, H. Tatlidil, and N. Maksayi (2017). Forecasting value-atrisk with two-step method: Garch-exponentiated odd log-logistic normal model. Romanian Journal of Economics Forecasting 20(4), 97–115. Altun, E. and G. M. Cordeiro (2019). The unit-improved second-degree lindley distribution: inference and regression modeling. Computational Statistics 35(1), 259–279. Altun, E., H. M. Yousof, S. Chakraborty, and L. Handique (2018). Zografos-balakrishnan burr XII distribution: regression modeling and applications. International Journal of Mathematics and Statistics 19(3), 46–70. Altun, E., H. M. Yousof, and G. G. Hamedani (2018). A new generalization of generalized half-normal distribution: properties and regression models. JSDA 5(7), 1–16. Alzaatreh, A., M. A. Aljarrah, M. Smithson, S. H. Shahbaz, M. Q. Shahbaz, F. Famoye, and C. Lee (2020). Truncated family of distributions with applications to time and cost to start a business. Methodology and Computing in Applied Probability, 1–23. Amini, M., S. M. T. K. Mirmostafaee, and J. Ahmadi (2012). Log-gamma-generated families of distributions. Statistics, iFirst 48(4), 913–932. Ansari, S. I. and H. Eledum (2018). An extension of pareto distribution. Journal of Statistics Applications & Probability 7(3), 443–455. Ansari, S. I. and Z. M. Nofal (2020). The lomax exponentiated weibull model. Japanese Journal of Statistics and Data Science, 1–19. Ansari, S. I., H. Rezk, and H. M. Yousof (2020). A new compound version of the generalized lomax distribution for modeling failure and service times. PJSOR 16(1), 95–107. Ansari, S. I., M. H. Samuh, and A. Bazyari (2019). Cubic transmuted power function distribution. Gazi University Journal of Science 32(4), 1322–1337.

References

427

Anzarra, L., S. Sarpong, and S. Nasiru (2020). Chen-g class of distributions. Cogent Mathematics & Statistics 7(1). Apam, B., N. Suleman, and E. Adjei (2019). Lomax weibull distribution. asian journal of probability and statistics (ajpas). 5(3), 1–18. Arifa, S., M. Z. Yab, and A. Ali (2017). The modified burr III g family of distributions. JDS 15(1), 41–60. Arshad, R, M. I., C. Chesneau, and F. Jamal (2019). The odd gamma weibull-geometric model: theory and applications. Mathematics (MDPI) 7(399), 1–18. Arshad, M. R. I., C. Chesneau, S. Ghazali, F. Jamal, and M. Mansoor (2019). The gamma power half-logistics distribution: theory and applications. HAL Id: hal-02306651; hal.archives-ouvertes.fr/hal-02306651, 1–19. Arshad, M. Z., M. Z. Iqbal, and M. Ahmad (2020). Exponentiated power function distribution: properties and applications. JSTA 19(2), 297–313. Aryal, G. G. (2019). On the beta exponential pareto distribution. Statistics, Optimization & Information Computing 7(2), 417–438. Aryal, G. R., K. P. Pokhrel, N. Khanal, and C. P. Tsokos (2019). Reliability models using the composite generalizers of weibull distribution. AODS, 23. Aryuyuen, S. and W. Bodhisuwan (2020). The type II topp-leone-power lomax distribution with analysis in lifetime data. Journal of Statistical Theory and Practice 14(31), 1–19. Asgharzadeh, A., M. Bourguignon, and M. Ghorbanzadeh (2016). The generalized inverse nadarajah-haghigh distribution. Journal of Statistics Applications and Probability. Aslam, M., Z. Asghar, Z. Hussain, and S. F. Shah (2020). A modified t-x family of distributions: classical and Bayesian analysis. Journal of Taibah University for Science 14(1), 254–264. Aslam, M., Z. Hussain, and Z. Asghar (2019). Modified generalized marshall-olkin family of distributions. International Journal of Advanced Statistics and Probability (JASP) 7(1), 18–27. Ateeq, K., T. B. Qasim, and A. R. Alvi (2020). An extension of rayleigh distribution and applications. Cogent Mathematics & Statistics (forthcoming). 6(1). Awodutire, P. O., E. C. Nduka, and M. A. Ijomah (2020). The beta type i generalized half logistic distribution: properties and applications. Asian Journal of Probability and Statistics (AJPAS) 6(2), 27–41. Badamasi, A. and V. V. Singh (2018). New odd generalized exponential-exponential distribution: its properties and application. Biostatistics & Biometrics 6(3), 1–6. Badr, M. M. (2019). Beta generalized exponentiated fréchet distribution with applications. Open Physics 17(1), 687–697.

428

References

Badr, M. M., I. Elbatal, F. Jamal, C. Chesneau, and M. Elgarhy (2020). The transmuted odd fréchet-g family of distributions: theory and applications. Mathematics 8(958), 1–20. Baharith, L. A., K. M. AL-Beladi, and H. S. Klakattawi (2020). The odd exponential-pareto iv distribution: regression model and application. Entropy 22(5), 497. Bakouch, H., C. Chesneau, and J. Leao (2018). A new lifetime model with a periodic hazard rate and application. Journal of Statistical Computation and Simulation 88(11), 2048–2065. Bakouch, H., F. Jamal, C. Chesneau, and A. Nasir (2017). A new transmuted family of distributions: properties and estimation with applications. hal-01570370; hal.archivesouvertes.fr/hal-01570370v3, 1–21. Bakouch, H., M. N. Khan, T. Hussain, and C. Chesneau (2018). A power log-dagum distribution: estimation and applications. Journal of Applied Statistics 46(5), 874–892. Bakouch, H. S., A. Saboor, and M. N. Khan (2020). Modified beta linear exponential distribution with hydrologic applications. Annals of Data Science, 1–27. Balaswamy, S. (2018). Transmuted half normal distribution. International Journal of Scientific Research in Mathematical and Statistical Sciences 5(4), 163–170. Bantan, R. A. R., A. S. Hassan, and M. Elsehetry (2020). Generalized marshallolkin inverse lindley distribution with applications. Computers, Materials & Continua (CMC) 64(3), 1505–1526. Bantan, R. A. R., A. S. Hassan, M. Elsehetry, and B. M. G. Kibria (2020). Half-logistic xgamma distribution: properties and estimation under censored samples. Discrete in Nature and Society vol. 2020, 1–18. Bantan, R. A. R., F. Jamal, C. Chesneau, and M. Elgarhy (2019a). A new power topp-leone generated family of distributions with applications. Entropy 21(12), 1177. Bantan, R. A. R., F. Jamal, C. Chesneau, and M. Elgarhy (2019b). Truncated inverted kumaraswamy generated family of distributions with applications. Entropy 21(11), 1– 22. Bantan, R. A. R., F. Jamal, C. Chesneau, and M. Elgarhy (2020a). On a new result on the ratio exponentiated general family of distributions with applications. Mathematics (MDPI) 8(598), 1–20. Bantan, R. A. R., F. Jamal, C. Chesneau, and M. Elgarhy (2020b). Type II power toppleone generated family of distributions with statistical inference and applications. Symmetry 12(75), 1–24. Barnard, R. W., C. Perera, J. G. Surles, and A. A. Trindade (2019). The linearly decreasing stress weibull (ldsweibull): a new weibull-loke distribution. Journal of Statistical Distributions and Applications 6(1), 1–21.

References

429

Barriga, G. D. C., G. M. Cordeiro, D. K. Dey, V. G. Cancho, F. Louzada, and A. K. Suzuki (2018). The marshall-olkin generated gamma distribution. Communications for Statistical Applications and Methods 25(3), 245–261. Basalamah, D., W. Ning, and A. Gupta (2018). The beta skew-t distribution and its properties. Journal of Statistical Theory and Practice 12(4), 837–860. Basheer, A. (2020). Marshall-olkin alpha power inverse exponential distribution: properties and applications. Annals of data science, 1–13. Basheer, A. M. (2019). Alpha power inverse weibull distribution with reliability application. Journal of Taibah University for Science 13(1), 423–432. Beghriche, A. and H. Zeghdoudi (2019). A size biased gamma lindley distribution. Thailand Statistician 17(2), 179–189. Bhat, A. A. and S. P. Ahmad (2020). A new generalization of rayleigh distribution: properties and applications. Pak. Journal Statist. 36(3), 225–250. Bhati, D., E. Calderin-Ojeda, and M. Meenakshi (2019). A new heavy tailed class of distributions which includes the pareto. Risls (MDPI) 7(4), 99. Bleed, S. O. (2020). Four parameters kumarswamy reciprocal family of distributions. JDS 18(1), 101–114. Bleed, S. O. and A. E. A. Abdelali (2018). Transmuted arcsine distribution: properties and application. International Journal of Research-Granthaalayah 6(10), 38–47. Borguignon, M., I. Ghosh, and G. M. Cordeiro (2016). General results for the transmuted family of distributions and new models. Journal of Probability and Statistics 2016, 1–12. Boshi, M. A. A., S. H. Abid, and N. H. Al-Noor (2019). Generalized gamma-generalized gompertz distribution. iop conf. Journal of Physics: Conference Series 1591(1), 1–15. Bouali, D. L., C. Chesneau, V. K. Sharma, and H. S. Bakouch (2019). A new class of distributions as a finite functional mixture using functional weights. (Personal Communication). Bourguignon, M., R. B. Silva, and G. Cordeiro (2014). The weibull-g family of probability distributions. Journal of Data Science 12(1), 53–68. Brito, E., G. M. Cordeiro, H. M. Yousof, M. Alizadeh, and G. O. Silva (2017). The toppleone odd log-logistic family of distributions, j. Statistical Computation and Simulation (JSCS) 87(15), 3040–3058. Bukoye, A. and G. M. Oyeyemi (2018). on development of four-parameters exponentiated generalized exponential distribution. Pak.Journal Statist. 34(4), 331–358. Cakmakyapan, S. and G. Ozel (2018). Lindley-rayleigh distribution with application to lifetime data. Journal of Reliability and Statistical Stydies 11(2), 9–24.

430

References

Castellars, F. and A. Lemonte (2015). A new generalized weibull generated by gamma random variables. Journal of the Egyptian Mathematical Society 23(2), 382–390. Castellars, F. and A. Lemonte (2016). On the gamma dual weibull model. American Journal of Mathematical and Management Sciences 35(2), 124–132. Celik, N. (2018). Some cubic rank transmuted distributions. JAMSI 14(2), 27–43. Chakrabarty, J. B. and S. Chowdhury (2019). Compounded inverse weibull distributions: properties, inference and applications. Communications in Statistics-Simulation and Computation 48(7), 2012–2033. Chakraborty, S., L. Handique, E. Altun, and H. M. Yousof (2019). A new statistical model for extreme values: properties and applications. Int. Journal Open Problems Compt. Math 12(1), 67–85. Chakraborty, S., L. Handique, and F. Jamal (2020). The kumarswamy poisson-g family of distributions: its properties and applications. Annals of Data Science, 1–19. Chakraborty, S., L. Handique, and R. M. Usman (2020). A simple extension of burr-III distribution and its advantages over existing ones in modeling failure time data. AODS 7(1), 17–31. Chakraborty, S. and G. C. Sarma (2017). Doubly truncated extreme value distribution of type I. International Journal of Agricultural and Statistical Sciences 13(2), 379–386. Chaves, N. L., C. L. N. Azevedo, F. Vilca-Labra, and J. S. Norbe (2019). A new birnbaumsaunders type distribution based on the skew-normal model under a centered parameterization. Chilean Journal of Statistics 10(1), 55–76. Chesneau, C., H. Bakouch, and V. K. Sharma (2018). A new family of distributions based on a poly-exponential transformation. Journal of Testing and Evaluation 48(1), 289–307. Chesneau, C., H. S. Bakouch, and T. Hussain (2019). A new class of probability distributions via cosine and sine functions with applications. Commun. Statist. Simulation and Computation 48(8), 2287–2300. Chesneau, C., H. S. Bakouch, and M. N. Khan (2020). A weighted transmuted exponential distribution with environmental applications. Statistics, Optimization & Information Computing. Chesneau, C. and S. Djibrila (2019). The generalized odd inverted exponential-g family of distributions: properties and applications. Eurasian Bulletin of Mathematics (EBM) 2(3), 86–110. Chesneau, C. and T. El Achi (2020). Modified odd weibull family of distributions: properties and applications. hal.archives-ouvertes.fr/hal-02317235. 1-21. Chesneau, C. and F. Jamal (2019). The sine kumarswamy-g family of distributions. Journal of Mathematical Extension 20, 20.

References

431

Chesneau, C., L. Tomy, J. Gillariose, and F. Jamal (2020). The inverted modified lindley distribution. Journal of Statistical Theory and Practice (JSTP) 14(3), 1–17. Chinazom, O. C., N. E. Chinaka, and I. M. Azubuike (2019). The marshall-olkin extended weibull-exponential distribution: properties and applications. Journal of Asian Scientific Research 9(10), 158–172. Chipepa, F., B. O. Oluyede, and B. Makubate (2019). A new generalized family of odd lindley-g distributions with applications. International Journal of Statistics and Probability (IJSP) 8(6), 1–23. Chipepa, F., B. O. Oluyede, and B. Makubate (2020). The topp-leone marshall-olkin-g family of distributions with applications. International Journal of Statistics and Probability (IJSP) 9(4), 15–32. Chipepa, F., B. O. Oluyede, B. Makubate, and A. F. Fagbamigbe (2019). The beta odd lindley-g family of distributions with applications. Journal of Probability and Statistical Science 17(1), 51–84. Cordeiro, G. M., A. Z. Afify, E. M. M. Ortega, A. K. Suzuki, and M. E. Mead (2019). The odd lomax generator of distributions: properties, estimation and applications. Journal of Computational and Applied Mathematics 351, 41–53. Cordeiro, G. M., A. Z. Afify, H. M. Yousof, S. Cakmakyapan, and G. Ozel (2018). The lindley weibull distribution: properties and applications. Anais Da Academia Brasileira de Ciencias (forthcoming) 90(3), 2579–2598. Cordeiro, G. M., M. Alizadeh, and P. R. D. Marinho (2015). The type i half-logistic family of distributions. Journal of Statistical Computation and Simulation 86(4), 707–728. Cordeiro, G. M., M. Alizadeh, E. M. M. Ortega, and L. H. V. Serrano (2016). The zografos-balakrishnan odd log-logistic family of distributions: properties and applications. hacettepe j. of mathematics and statistics (hijms). 45, 1781–1803. Cordeiro, G. M., M. Alizadeh, G. Ozel, B. Hosseini, E. M. M. Ortega, and E. Altun (2016). The generalized odd log-logistic family of distributions: properties, regression models and applications. Journal of Statistical Computation and Simulation 87(5), 908–932. Cordeiro, G. M., E. Altun, M. C. Korkmaz, R. R. Pescim, A. Z. Afify, and H. M. Yousof (2020). The xgamma family: censored regression modeling and applications. RevstatStatistics Journal 18(5), 593–612. Cordeiro, G. M., R. J. Cintra, L. C. Rego, and A. D. C. Nascimento (2019). The gamma generalized normal distribution: A descriptor of sar imagery. Journal of Computational and Applied Mathematics 347, 257–272. Cordeiro, G. M., M. C. S. Lima, A. Z. Gomes, C. O. da Silva, and E. M. M. Ortega (2016). The gamma extended weibull distribution. Journal Statistical Distributions and Applications 3, 1–18.

432

References

Cordeiro, G. M., M. C. S. Lima, E. M. M. Ortega, and A. K. Suzuki (2018). A new extended birnbaum-saunders model: properties, regression and applications. STATS 1(1), 32–47. Cordeiro, G. M., M. Mansoor, and S. B. Provost (2019). The harris extended lindley distribution for modeling hydrological data. Chilean Journal of Statistics 10(1), 77–94. Cordeiro, G. M., E. M. M. Ortega, and D. C. C. da Cunha (2013). The exponentiated generalized class of distributions. Journal of Data Science 11, 1–27. Cordeiro, G. M., E. M. M. Ortega, and S. Nadarajah (2010). The kumaraswamy weibull distribution with application to failure data. Journal Franklin Institute 347, 1399–1429. Cordeiro, G. M., F. Prataviera, M. D. C. S. Lima, and E. M. M. Ortega (2019). The marshallolkin extended flexible weibull regression model for censored lifetime data. Model Assisted Statistics and Applications 14, 1–17. Cortés, M. A., D. Elal-Olivero, and J. F. Olivares-Pacheco (2018). A new class of distributions generated by the extended bimodal-normal distribution. J. of Probability and Statistics 2018, 1–10. da Silva Braga, O. A., and d. C. E. M. M. (2016). The odd log-logistic normal distribution: theory and applications in analysis of experiments. Journal of Statistical Theory and Practice 10(2), 311–335. Dey, S., A. Alzaatreh, C. Zhang, and D. Kumar (2017). A new extension of generalized exponential distribution with application to ozone data. Ozone: Science & Engineering 39(4), 273–285. Dey, S., I. Ghosh, and D. Kumar (2019). Alpha-power transformed lindley distribution: properties and associated inference with application to earthquake data. Annals of Data Science 6(4), 623–650. Dey, S., M. Nassar, and D. Kumar (2018). Alpha power transformed inverse lindley distribution: a distribution with an upside-down bathtub hazard function. Journal of Computational and Applied Mathematics 348, 130–145. Dey, S., M. Nassar, D. Kumar, A. Alzaatreh, and M. H. Tahir (2019). A new lifetime distribution with decreasing and upside-down bathtub-shaped hazard rate function. STATISTICA 79(4), 399–426. Dias, C., G. Cordeiro, M. Alizadeh, P. Diniz Marinho, and H. Campos Coêlho (2016). Exponentiated marshall-olkin family of distributions. Journal of Statistical Distributions and Applications 3:15, 1–21. Domma, F., A. Eftekharian, A. Z. Afify, M. Alizadeh, and I. Ghosh (2018). The odd log-logistic dagym distribution: properties and applications. Revista Colombiana de Estadistica 41(1), 109–135. Doostmoradi, A. (2018). A new distribution with two parameters to lifetime data. Biostatistics and Biometrics Journal (JP Juniper) 8(2), 1–6.

References

433

Efe-Eyefia, E., J. T. Eghwerido, and S. C. Zelibe (2020). Theoretical analysis of the weibull alpha power inverted exponential distribution: properties and applications. Gazi University Journal of Science 33(1), 265–277. Eghwerido, J. T., E. Efe-Eyefia, and C. S. Zelibe (2020). The transmuted alpha power-g family of distributions. Research Gate. Eghwerido, J. T., L. C. Nzei, I. J. David, and O. D. Adubisi (2020). The gompertz extended generalized exponential distribution: properties and applications. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69(1), 739– 753. Eghwerido, J. T., S. C. Zelibe, and E. Efe-Eyefia (2020). Gompertz-alpha power inverted exponential distribution: properties and applications. Thailand Statistician 18(3), 319– 332. Eissa, F. H. (2017). The exponentiated kumaraswamy-weibull distribution with application to real data. Int J Stat Probab 6(6), 167–182. Ekhosuehi, N., L. C. Nzei, and F. Opone (2020). A new mixture of exponential-gamma distribution. Gazi University Journal of Science 33(2), 548–564. Ekhosuehi, N. and F. Opone (2018). A three parameter generalized lindley distribution: properties and application. Statistica 78(3), 233–249. Ekum, M. I., M. O. Adamu, and E. E. Akarawak (2020). T-dagum: A way of generalizing dagum distribution using lomax quantile function. Journal of Probability and Statistics, Article ID 1641207 2020, 1–17. El-Damcese, M. A., A. Mustafa, and M. S. Eliwa (2016). Exponentiated generalized weibull-gompertz distribution. (Personal communication). El-Sherpieny, E. and M. Elsehetry (2019). Kumaraswamy type i half logistic family of distributions with applications. GU J Sci 32(1), 333–349. Elbatal, I., A., M. Z., Elgarhy, and A. Almarashi (2019). A new alpha power transformed family of distributions: properties and applications to the weibull model. Journal of Nonlinear Science and Applications 12(1), 1–20. Elbatal, I. and H. Al-Mofleh (2019). Extended marshall-olkin modified weibull distribution: properties and applications. (Personal Communication). Elbatal, I., A. M. Almarashi, M. Elgarhy, and M. Ahsan ul Haq (2018). Type I half logistic power lindley distribution with applications. far East Journal of Mathematical Sciences (FJMS), 1–19. Elbatal, I., E. Altun, A. Z. Afify, and G. Ozel (2019). The generalized burr XII power series distributions with properties and applications. AODS 6(3), 571–597. Elbatal, I., L. S. Diab, and N. A. Abdul Alim (2013). Transmuted generalized linear exponential distribution. International Journal of Computer Applications 83(17), 29–37.

434

References

Elbatal, I., F. Jamal, C. Chesneau, M. Elgarhy, and S. Alrajhi (2019). The modified beta gompertz distribution: theory and applications. MDPI Mathematics 7(3), 1–17. Elbatal, I. and M. G. Khalil (2019). A new extension of lindley geometric distribution and its applications. PJSOR 15(2), 249–263. Elbatal, I., F. Louzada, and D. C. T. Granzotto (2018). A new lifetime model: the kumaraswamy extension exponential distribution. Biostatistics & Bioinforrmatics 2(1), 1– 9. Elbiely, M. M. (2019). The dual exponentiated weibull model. Journal of Mathematics and Statistics 15, 122–135. Elbiely, M. M. and H. M. Yousof (2019). A new inverse weibull distribution: properties and applications. Journal of Mathematics and Statistics 15(1), 30–43. Eledum, H. (2020). Some cubic transmuted exponentiated pareto-1 distribution. Journal of Mathematics and Statistics 16, 113–124. Elgarhy, M. and S. Alrajhi (2019). The odd fréchet inverse rayleigh distribution: statistical properties and applications. Journal of Nonlinear Sci. Appl 12(5), 291–299. Elgarhy, M., M. Ashan ul Haq, and I. Perveen (2019). Type II half logistics exponential distribution with application. Annals of Data Science (AODS) 5(2), 245–257. Elgarhy, M., A. Nasir, F. Jamal, and G. Ozel (2018). The type II topp-leone generated family of distributions: properties and applications. Journal of Discrete Mathematical Sciences & Cryptography 21(8), 1529–1551. Elgarhy, M., V. K. Sharma, and I. Elbatal (2018). Transmuted kumarswamy lindley distribution with application. Journal of Statistics & Management System 21(6), 1083–1104. Eliwa, M. S., M. El-Morshedy, and S. Ali (2020). Exponentiated odd chen-g family of distributions: statistical properties, Bayesian and non-bayesian estimation with applications. Journal of Applied Statistics, 1–27. Eliwa, M. S., M. El-Morshedy, and M. Ibrahim (2019). Inverse gompertz distribution: properties and different estimation methods with application to complete and censored data. Annals of Data Science 6(2), 321–339. Eliwa, M. S., M. Salah, Z. A. Alhussain, E. A. Ahmed, E. Altun, H. H. Ahmed, and M. ElMorshedy (2020). A new one-parameter lifetime distribution and its regression model with applications. Personal Communication). Elsayed, H. A. H. and H. M. Yousof (2019). The burr x nadarajah haghighi distribution: statistical properties and application to the exceedances of flood peaks data. Journal of Mathematics and Statistics 15, 146–157. Elsayed, H. A. H. and H. M. Yousof (2020a). Extended poisson generalized burr XII distribution. Journal of Applied Probability and Statistics (forthcoming).

References

435

Elsayed, H. A. H. and H. M. Yousof (2020b). The generalized odd generalized exponential fréchet model: univariate, bivariate and multivariate extensions with properties and applications to the univariate version. Pakistan Journal of Statistics and Operation Research, 529–544. Elsherpieny, E. A., Y. Y. Abdelall, and A. A. Mohamed (2017). On the exponentiated new weighted weibull distribution. International Journal of Engineering and Applied Sciences (IJEAS) 4(10), 41–50. Elsherpieny, E. A. and M. M. Elseherty (2019). Type II kumarswamy half logistic family of distributions with applications to exponential model. Annals of Data Science 6(1), 1–20. Eraikhuemen, I. B., A. F. Chama, A. I. Asongo, B. S. Yakura, and A. H. Bala (2020). Properties and applications of a transmuted power gompertz distribution. Asian Journal of Probability and Statistics (AJPAS) 7(1), 41–58. Eraikhuemen, I. B., T. G. Ieren, T. M. Mabur, M. Sa’ad, S. Kuje, and A. F. Chama (2019). A study on properties and applications of a lomax gompertz-makeham distribution. Asian Research Journal of Mathematics (ARJOM) 15(4), 1–27. Esbond, G. I. and S. W. O. Fuumilayo (2018). The exponentiated-epsilon distribution: its properties and applications. International Journal of Science and Research (IJSR). Esmaeili, H., E. Altun, F. Lak, and M. Alizadeh (2020). An extended odd log-logistic family of distributions: properties regression models and applications. (Personal Communication). Esmaeili, H., F. Lak, and M. E. Alizadeh, M.and Dehghan Monfared (2020). The alphabeta skew logistic distribution: properties and applications. Statistics Optimization and Information Computing 8(1), 304–317. Esmaeili, H., F. Lak, and E. Altun (2020). The ristic-balakrishnan odd log-logistic family of distributions: properties and applications. Statistics, Optimization and Information Computing 8(1), 17–35. Eyob, T. and R. Shanker (2018). A two-parameter weighted garima distribution with properties and application. Biometrics & Biostatistics International Journal 7, 234–242. Falgore, J. Y. (2020). The zubair-inverse lomax distribution with applications. Asian Journal of Probability and Statistics 8(3), 1–14. Falgore, J. Y. and S. I. Doguwa (2020). The inverse lomax-g family with application to breaking strength data. AJPAS 8(2), 49–60. Falgore, J. Y., S. I. Doguwa, and I. Audu (2019a). The properties of weibull-inverse lomax distribution. Falgore, J. Y., S. I. Doguwa, and I. Audu (2019b). The weibull-inverse lomax (wil) distribution with application on bladder cancer. Biometrics & Biostatistics International Journal (BBIJ) 8(5), 195–202.

436

References

Fatima, K. and S. P. Ahmad (2018). Characterization and Bayesian inference for exponentiated generalized standard inverse exponential distribution. Pakistan Journal of Statistics 34(4), 269–296. Fatima, K., S. Naqash, and S. P. Ahmad (2018). Exponentiated generalized inverse rayleigh distribution with applications in medical sciences. Pak.Journal Statist. 34(5), 425–439. Fayomi, A. (2019). The odd fréchet inverse weibull distribution with application. Journal of Nonlinear Sci. Appl 12(3), 165–172. Fernández, L. M. Z. and T. A. N. de Andrade (2020). The erf-g family: new unconditioned and log-linear regression models. Chilean Journal of Statistics 11(1), 3–23. Ferreira, C. S., V. H. Lachos, and A. M. Garay (2020). Inference and diagnostics for heteroscedastic nonlinear regression models under skew scale mixtures of normal distributions. Journal of Applied Statistics 47(9), 1690–1719. George, R. and S. Thobias (2019). Kumaraswamy marshall-olkin exponential distribution. Communications in Statistics-Theory and methods 48(8), 1920–1937. Gharaibeh, M. M. and A. I. Al-Omari (2019). Transmuted ishita distribution and its applications. Journal of Statistics Applications & Probability 8(2), 67–81. Ghitany, M. E., D. K. Al-Mutairi, N. Balakrishnan, and L. J. Al-Enezi (2013). Power lindley distribution and associated inference. Computational Statistics and Data Analysis 64, 20–33. Ghitany, M. E., J. Mazucheli, A. F. B. Menezes, and F. Alqallaf (2019). The unit-inverse gaussian distribution: a new alternative to two-parameter distributions on the unit interval. Communications in Statistics-Theory and Methods 48(14), 3423–3438. Ghosh, I. and A. Alzaatreh (2018). A new class of generalized logistic distribution. Communications in Statistics-Theory and Methods 47(9), 2043–2055. Ghosh, I., S. Dey, and D. Kumar (2019). Bounded m-o extended exponential distribution with applications. Stochastics and Quality Control 34(1), 35–51. Ghosh, I. and S. Nadarajah. On some further properties and application of weibull-r family of distributions. Annals of Data Science 5(3), 387–399. Ghosh, I. and H. K. T. Ng (2019). A class of skewed distributions with applications in environmental data. Communications in Statistics: Case Studies, Data Analysis and Applications 5(4), 346–365. G˝und˝uz, S. and M. C. Korkmaz (2020). A new unit distribution based on the unbounded johnson distribution : the unit johnson sU distribution. PJSOR 16(3), 471–490. Gillariose, J. and L. Tomy (2020). The marshall-olkin extended power lomax distribution with applications. Pakistan Journal of Statistics and Operation Research 16(2), 331–341.

References

437

Glänzel, W. (1987). A characterization theorem based on truncated moments and its application to some distribution families. Mathematical Statistics and Probability Theory, 75–84. Glänzel, W. (1990). Some consequences of a characterization theorem based on truncated moments. Statistics: A Journal of Theoretical and Applied Statistics 21(4), 613–618. Goldust, M., S. Rezaei, and S. Nadarajah (2017). Lifetime distributions motivated by series and parallel structures. Communications in Statistics-Simulation and Computation 48(2), 556–579. Gomes-Silva, F., T. A. N. de Andrade, and M. Bourguignon (2018). Risti´c-balakrishnan extended exponential distribution. Acta Scientiarum Technology 40. Gomes-Silva, F., A. Percontini, E. de Brito, M. W. Ramos, R. Venâncio, and G. M. Cordeiro (2017). The odd lindley-g family of distributions. Austrian Journal of Statistics 46(1), 57–79. Gómez, Y. M., H. Bolfarine, and H. W. Gómez (2014). A new extension of the exponential distribution. Revista Colombiana de Estadistica 37(1), 25–34. Goual, H. and H. M. Yousof (2019). Validation of burr XII inverse rayleigh model via a modified chi-square goodness-of-fit test. Journal of Applied Statistics 47(3), 393–423. Goual, H., H. M. Yousof, and M. M. Ali (2020). Lomax inverse weibull model: properties, applications and a modified chi-squared goodness-of-fit test for validation. Journal of Nonlinear Sciences & Applications (JNSA) 13(6), 330–353. Gui, W. (2013). A marshall-olkin power log-normal distribution and its applications to survival data. IJSP 2(1), 63–72. Habibi, M. and A. Asgharzadeh (2018). Power binomial exponential distribution: modeling, simulation and application. Commun. Statist. Simulation and Computation 47(10), 3042–3061. Hakamipour, N., S. Nadarajah, and S. Rezaei (2012). Logarithmic mixture distribution. (Personal communication). Hamed, M. S. (2018a). The burr XII exponentiated weibull model. Journal of Statistics and Applications 1(2), 15–31. Hamed, M. S. (2018b). The weibull generalized exponentiated weibull distribution: theory and applications. Journal of Statistics and Applications 1(2), 1–15. Hamed, M. S. (2020). Extended poisson-fréchet distribution: mathematical properties and applications to survival and repair times. JDS 18(2), 319–341. Hamed, M. S., F. Aldossary, and A. Z. Afify (2020). The four-parameter fréchet distribution: properties and applications. PJSOR 16(2), 249–264.

438

References

Hamedani, G. (2019). Characterizations of Recently Introduced Univariate Continuous Distributions II. Mathematics Research Developments Series. New York: Nova Science Publishers. Hamedani, G. and F. Safavimanesh (2017). Characterizations and infinite divisibility of certain 2016 univariate continuous distributions. International Mathematical Forum 12(5), 195–228. Hamedani, G. G. (2016). On characterizations and infinite divisibility of recently introduced distribution. SMH 53(4), 467–511. Hamedani, G. G. (2017). On characterizations of transmuted geometric-g family of distributions. Pakistan Journal of Statistics 33(2), 129–134. Hamedani, G. G. (2018). Characterizations and infinite divisibility of certain recently introduced distributions III. IJSP 7(1), 39–71. Hamedani, G. G. (2020). Generalized lindley distribution. (Preprint). Hamedani, G. G. and S. M. Najibi (2016). Certain characterizations of recently introduced distributions. PJSOR 12(4), 551–577. Hamedani, G. G. and M. I. Resti´c (2016). Moments of record values and characterizations of the marshall-olkin extended distribution. JSTA 15, 115–124. Hameldarbandi, M. and M. Yilmaz (2020). Some comments on methodology of cubic rank transmuted distributions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69(2), 167–176. Hanagal, D. D. and N. N. Bhalerao (2019). Modeling on generalized extended inverse weibull software reliability growth model. JDS 17(3), 573–590. Handique, L. Jamal, F. and S. Chakraborty (2020a). On a family that unifies generalized marshall-olkin and poisson-g family of distribution. (Personal Communication). Handique, L., M. Ahsan ul Haq, and S. Chakraborty (2020). Generalized modified exponential-g family of distributions: its properties and applications. International Journal of Mathematics and Statistics 21(1), 1–17. Handique, L. and S. Chakraborty (2017a). The beta generated kumaraswamy marshallolkin-g family of distributions with applications. International Journal of Agricultural and Statistical Sciences 13(2), 721–733. Handique, L. and S. Chakraborty (2017b). The new beta generalized marshall-olkin kumaraswamy-g family of distributions with applications. Malaysian Journal of Science 36(3), 157–174. Handique, L. and S. Chakraborty (2017c). A new family of distributions which unifies the kumaraswamy marshall-olkin and beta marshall-olkin family of distributions with applications. Academia. edu, 1–18.

References

439

Handique, L. and S. Chakraborty (2018). A new four-parameter extension of burr-XII distribution: its properties and applications. Japanese Journal of Statistics and Data Science 1(2), 271–296. Handique, L., S. Chakraborty, and T. A. N. de Andrade (2019). The exponentiated generalized marshall-olkin family of distributions: Its properties and applications. AODS 6(3), 391–411. Handique, L., S. Chakraborty, and F. Jamal (2020). Beta poisson-g family of distributions: its properties and application with failure data. (Forthcoming). Handique, L., U. R. and S. Chakraborty (2020b). New extended burr iii distribution: its properties and applications. Thailand Statistician 18(3), 267–280. Hashimoto, E. M., E. M. M. Ortega, G. M. Cordeiro, V. G. Cancho, and C. Klauberg (2018). Zero-spiked regression models generated by gamma random variables with application in the resin oil production. Journal of Statistical Computation and Simulation 89(1), 52–70. Hashmi, S. and U. R. Ahsan ul Haq, M. (2019). A generalized exponential distribution with increasing, decreasing and constant shape hazard rate. Electronic Journal of Applied Statistical Analysis 12(1), 223–244. Hashmi, S., U. R. Ahsan ul Haq, M., and G. Ozel (2016). The weibull-moment exponential distribution: properties, characterizations and applications. Journal of Reliability and Statistical Studies, 1–22. Hashmi, S. and H. Gull (2018). A new weibull-lomax (t-x) distribution & its applications. Pak. Journal Statist. 34(6), 495–512. Hassan, A., S. A. Wani, S. Shafi, and S. B. Ahmad (2020). Lindley-quasi xgamma distribution: properties and applications. Pak. Journal Statist. 36(1), 73–89. Hassan, A. S. and M. Abd-Allah (2018). Exponentiated weibull-lomax distribution: properties and estimation. JDS 16(2), 277–298. Hassan, A. S. and M. Abd-Allah (2019). AODS 5(2), 259–278.

On the inverse power lomax distribution.

Hassan, A. S., I. B. Abdul-Mpniem, and K. A. E. Gad (2020). A generalized transmuted moment exponential distribution: properties and application. Academic Journal of Applied Mathematical Sciences 6(5), 41–52. Hassan, A. S. and S. M. Assar (2020). A new class of power function distribution: properties and applications. Annals of Data Science, 1–21. Hassan, A. S. and M. Elgarhy (2019). Exponentiated weibull weibull distribution: statistical properties and applications. GU J Sci 32(2), 616–635. Hassan, A. S., M. Elgarhy, and Z. Ahmad (2019). Type II generalized topp-leone family of distributions: properties and applications. JDS 17(4), 638–659.

440

References

Hassan, A. S., M. Elgarhy, M. Ahsan ul Haq, and S. Alrajhi (2019). On type II half logistic weibull distribution with applications. Mathematical Theory and Modeling 9(1), 49–63. Hassan, A. S., M. Elgarhy, S. G. Nassr, Z. Ahmad, and S. Alrajhi (2019). Truncated weibull fréchet distribution: statistical inference and applications. Journal of Computational and Theoretical Nanoscience 16, 1–9. Hassan, A. S., M. Elgarhy, and M. Shakil (2017). Type II half logistic family of distributions with applications. Pak.Journal stat.oper.res XIII(2), 245–264. Hassan, A. S., E. Elshrpieny, and R. Mohamed (2019). Odd generalized exponential power function distribution: properties and applications. GU J Sci 32(1), 351–370. Hassan, A. S. and R. E. Mohamed (2019). Weibull inverse lomax distribution. PJSOR XV(3), 587–603. Hassan, A. S., R. E. Mohamed, M. Elgarhy, and A. Fayomi (2019). Alpha power transmuted extended exponential distribution: properties and applications. Journal of Nonlinear Sci. Appl 12(4), 239–251. Hassan, A. S. and S. G. Nassr (2018). The inverse weibull generator of distributions: properties and applications. JDS 16(4), 723–742. Hassan, A. S. and S. G. Nassr (2019a). A new generalization of power function distribution: properties and estimation based on censored samples. Thailand Statistician 18(2), 215– 234. Hassan, A. S. and S. G. Nassr (2019b). Power lindley-g family of distributions. AODS 6(2), 189–210. Hassan, A. S., M. A. H. Sabry, and A. M. Elsehetry (2020). Truncated power lomax distribution with application to flood data. Journal Stat. Appl. Pro 9(2), 347–359. Hassan, N. J., A. H. Nassar, and J. M. Hadad (2020). Distributions of the ratio and product of two independent weibull and lindley random variables. International Journal of Probability and Statistics, Article ID: 5693129, 1–8. Hazarika, P. J., S. Shah, and S. Chakraborty (2019). Balakrishnan alpha skew normal distribution: properties and applications. Technical report, Preprint, 21 pages. He, W., Z. Ahmad, A. Z. Afify, and H. Goual (2020). The arcsine exponentiated-x family: validation and insurance application. Complexity, 8394815:1-8394815:18. Hemeda, S. E. and A. M. Abdallah (2020). Sinh inverted exponential distribution: simulation & application to neck cancer disease. IJSP 9(5), 11–22. Huo, X., S. K. Khosa, Z. Ahmad, Z. Almaspoor, M. Ilyas, and M. Aamir (2020). A new lifetime exponential-x family of distributions with applications to reliability data. Mathematical Problems in Engineering 2020, 1–16. Article ID 1316345.

References

441

Hussain, T., H. S. Bakouch, and C. Chesneau (2020). A new probability model with application to heavy-tailed hydrological data. Environmental and Ecological Statistics 26(2), 127–151. Hussain, Z., M. Aslam, and Z. Asghar (2019). On exponentiated negative-binomial-x family of distributions. AODS. Hutson, A., T. L. Mashtare Jr, and G. S. Mudholkar (2020). Log-epsilon-skew normal: a generalization of the log-normal distribution. Communications in Statistics-Theory and Methods 49(17), 4197–4215. Ibrahim, M. (2019). A new extended fréchet distribution: properties and estimation. PJSOR XV(3), 773–796. Ibrahim, M. (2020). The compound poisson rayleigh burr XII distribution: properties and applications. Journal of Applied Probability and Statistics 15(1), 73–97. Ibrahim, M., E. Altun, and H. M. Yousof (2020). A new distribution for modeling lifetime data with different methods of estimation and censored regression modeling. Statistics, Optimization and Information Computing 8, 610–630. Ibrahim, M. and H. M. Yousof (2020). Transmuted topp-leone weibull lifetime distribution: statistical properties and different method of estimation. PJSOR 16(3), 501–515. Ibrahim, N. A. and M. A. Khaleel (2020). Generalizations of burr type x distribution with applications. ASM Sc 13, 1–8. Ibrahim, S., B. O. Akanji, and L. H. Olanrewaju (2020). On the extended generalized inverse exponential distribution with its applications. Asian Journal of Probability and Statistics (AJPAS) 7(3), 14–27. Idemudia, R. and N. Ekhosuehi (2019). A new three parameter paralogistic distribution: its properties and applications. JDS 17(2), 239–258. Ieren, T. G., S. S. Abdulkadir, and A. A. Issa (2020). Odd lindley-rayleigh distribution: its properties and applications to simulated and real life datasets. Journal of Advances in Mathematics and Computer Science 35(1), 63–88. Ieren, T. G. and J. Abdullahi (2020). Properties and applications of a two-parameter inverse exponential distribution with a decreasing failure rate. Pak. Journal Statist. 36(3), 183– 206. Ieren, T. G., F. M. Kromtit, B. U. Agbor, I. B. Eraikhuemen, and P. O. Koleoso (2019). A power gompertz distribution: model, properties and application to bladder cancer data. Asian Research Journal of Mathematics, 1–14. Ihtisham, S., A. Khalil, S. Manzoor, S. A. Khan, and A. Ali (2019). Alpha-power pareto distribution: Its properties and applications. PloS One 14(6), 1–15. Ijaz, M. and S. M. A. Alamgir (2019). Lomax exponential distribution with an application to real-life data. PLOS ONE 14(12), 1–16.

442

References

Ijaz, M., S. M. Asim, A. M. Farooq, S. A. Khan, and S. Manzoor (2020). A gull alpha power weibull distribution with applications to real and simulated data. PLOS ONE; 15(6). Ikechukwu, A. F., R. F. Emmanuel, and J. T. Eghwerido (2020). The type II topp-leone generalized power ishita distribution with properties and applications. Thailand Statistician (forthcoming). Iqbal, M. Z., N. Ali, A. Razaq, T. Hussain, and M. Salman (2020). On generalized moment exponential distribution and power series distribution. Asian Journal of Probability and Statistics (AJPAS) 6(1), 1–21. Iqbal, M. Z., M. Z. Arshad, M. Ahmad, I. Ahmad, T. Iqbal, and M. A. Bhatti (2019). Double truncated transmuted fréchet distribution: properties and applications. Mathematical Theory and Modeling 9(3), 11–34. Iqbal, M. Z., M. M. Tahir, N. Riaz, S. A. Ali, and M. Ahmad (2017). Generalized inverted kumaraswamy distribution: properties and application. Open Journal of Statistics 7(4), 645–662. Iriate, Y. A. and M. A. Rojas (2019). Slashed power-lindley distribution. Communications in Statistics-Theory and Methods 48(7), 1709–1720. Iriate, Y. A., H. Varela, H. J. Gomez, and H. W. Gomez (2020). A gamma-type distribution with applications. Symmetry (MDPI) 12(5), 15. Jahanshahi, S. M. A., H. M. Yousof, and V. K. Sharma (2019). The burr x fréchet model for extreme values: mathematical properties, classical inference and Bayesian analysis. Pakistan Journal of Statistics and Operation Research 15(3), 797–818. Jamal, F., A. Abuzaid, M. Nasir, A. Saboor, and M. Khan (2018). New modified burr III distribution, properties and applications. HAL Id: hal-01902854; hal.archivesouvertes.fr/hal-01902854, 1–20. Jamal, F., H. Bakouch, and A. Nasir (2019). Odd burr III g-negative binomial family with application. Journal of Testing and Evaluation 49(5). Jamal, F., H. Bakouch, and A. Nasir (2020). A truncated general-g class of distributions with application to truncated burr-g family. (Personal Communication). Jamal, F. and C. Chesneau (2019). Box-cox gamma-g family of distributions: theory and applications. HAL Id: hal-02022458; hal.archives-ouvertes.fr/hal-02022458, 1–27. Jamal, F., C. Chesneau, and M. Elgarhy (2020). Type II general inverse exponential family of distributions. Journal of Statistics and Management Systems 23(3), 617–641. Jamal, F., I. Elbatal, C. Chesneau, M. Elgarhy, and A. S. Hassan (2019). Modified beta generalized linear failure rate distribution: theory and applications. Journal of Prime Research in Mathematics 15, 21–48. Jamal, F. and M. A. Nasir (2019). Some new members of the t-x family of distributions. hal.archives-ouvertes.fr/hal-01965176v3, 1–6.

References

443

Jamal, F., M. A. Nasir, G. Ozel, M. Elgarhy, and N. M. Khan (2019). Generalized inverted kumarswamy generated family of distributions: theory and applications. Journal of Applied Statistics 46(16), 2927–2944. Jamal, F., M. A. Nasir, M. H. Tahir, and N. H. Montazeri (2017). The odd burr-III family of distributions. Journal of Statistics Applications and Probability 6(1), 105–122. Jamal, F., H. Reyad, C. Chesneau, M. A. Nasir, and S. Othman (2019). The marshall-olkin odd lindley-g family of distributions: theory and applications. Punjab University Journal of Mathematics 51(7), 111–125. Jamal, F., H. M. Reyad, S. O. Ahmed, and M. A. A. Shah (2020). Mathematical properties and applications of minimum gumbel burr distribution. NED University Journal of Research-Applied Sciences 17(2), 1–14. Jamal, F., H. M. Reyad, S. O. Ahmed, M. A. A. Shah, and E. Altun (2019). Exponentiated half-logistic lomax distribution with properties and application. NED University Journal of Research-Applied Sciences 16(2), 1–11. Jamal, F., H. M. Reyad, M. A. Nasir, C. Chesneau, M. A. A. Shah, and S. O. Ahmed (2019). Topp-leone weibull-lomax distribution: properties, regression model and applications. HAL Id: hal-02270561; hal.archives-ouvertes.fr/hal-02270561, 1–24. Jamal, F., M. H. Tahir, M. Alizadeh, and M. A. Nasir (2017). On marshall-olkin burr x family of distributions. Tbilisi Journal of Mathematics 10(4), 175–199. Jan, R., N. Bashir, and T. R. Jan (2019). Alpha-power generalized inverse lindley distribution: properties and applications. JAPS 14(3), 117–130. Javad, M., T. Nawaz, and M. Irfan (2019). The marshall-olkin kappa distribution: properties and applications. Journal of King Saud University-Science 31(4), 684–691. Jayakumar, K. and K. K. Sankaran (2016). On a generalization of uniform distribution and its properties. Statistica 76(1), 83–91. Jayakumar, K. and K. K. Sankaran (2019). Discrete linnik weibull distribution. Communications in Statistics-Simulation and Computation 48(10), 3092–3117. Jayakumar, K. and K. K. Sankaran (2020a). Exponential intervened poisson distribution. Communications in Statistics-Theory and Methods, 1–31. Jayakumar, K. and K. K. Sankaran (2020b). A generalization of inverse marshall-olkin family of distributions. JDS 18(1), 1–43. Jodrá, P., F. W. Gómez, M. D. Jiménez-Gamero, and M. V. Alba-Gernández (2017). The power muth distribution. Mathematical Modeling and Analysis 22(2), 186–201. Joshi, S. and K. K. Jose (2020). Wrapped lindley distribution. Communications in StatisticsTheory and Methods 47(5), 1013–1021.

444

References

Kamal, R. M. and M. A. Ismail (2020). The flexible weibull extension-burr XII distribution: model, properties and applications. PJSOR 16(3), 447–460. Kehinde, O., A. Osebi, and D. Ganiyu (2018). A new class of generalized burr III distribution for lifetime data. International Journal of Statistical Distributions and Applications 4(1), 6–21. Khaleel, M. A., N. H. Al-Noor, and M. K. Abdal-Hameed (2020). Marshall-olkin exponential gompertz distribution: properties and applications. Periodicals of Engineering and Natural Sciences 8(1), 298–312. Khaleel, M. A., P. E. Oguntunde, M. T. Ahmed, N. A. Abrahim, and Y. F. Loh (2020). The gompertz flexible weibull distribution and its applications. Malaysian Journal of Mathematical Sciences 14(1), 169–190. Khaleel, M. A., P. E. Oguntunde, J. N. Al Abbasi, N. A. Ibrahim, and M. H. AbuJarad (2020). The marshall-olkin topp leone-g family of distributions: a family for generalizing probability models. Scientific African 8, 1–19. Khalil, A., M. Ijaz, K. Ali, W. K. Mahwani, M. Shafiq, P. Kuman, and W. Kuman (2020). A novel flexible additive weibull distribution with real-life applications. Communications in Statistics-Theory and Methods, 1–16. Khalil, M. G. (2019). A new distribution for modeling extreme values. JDS 17(3), 481–503. Khalil, M. G. and H. Rezk (2019). Extended poisson fréchet distribution and its applications. Pakistan Journal of Statistics and Operation Research XV(4), 905–919. Khan, M. N., A. Saeed, and A. Alzaatreh (2019). Weighted modified weibull distribution. Journal of Testing and Evaluation 47(5), 3751–3764. Khan, M. S. (2018). Transmuted generalized inverted exponential distribution with application to reliability data. Thailand Statistician 16(1), 14–25. Khan, M. S., R. King, and I. L. Hudson (2017). Transmuted new generalized inverse weibull distribution. Pakistan Journal of Statistics and Operation Research XIII(2), 227–296. Khan, M. S., R. King, and I. L. Hudson (2019a). Transmuted burr type x distribution with covariates regression modeling to analyze reliability data. American Journal of Mathematical and Management Sciences 39(2), 99–121. Khan, M. S., R. King, and I. L. Hudson (2019b). Transmuted exponentiated weibull distribution. Journal of Applied Probability and Statistics 14(2), 37–51. Kharazmi, O. and A. Saadatinik (2018). Hyperbolic sine-f families of distributions with an application to exponential distribution. Gazi University Journal of Science. Kharazmi, O., A. Saadatinik, and S. Jahangard (2019). Odd hyperbolic cosine exponentialexponential (ohc-ee) distribution. Annals of Data Science 6(4), 765–785.

References

445

Kharazmi, O., A. Saadatinik, and M. Tamandi (2019). A new continuous lifetime distribution and its application to the indemnity and aircraft windshield dataset. Mathematical Sciences and applications E-Notes (MSEAEN) 7(1), 102–112. Khosa, S. K., A. Z. Afify, Z. Ahmad, M. Zichuan, S. Hussain, and A. Iftikhar (2020). A new extended-f family: properties and applications to lifetime data. Journal of Mathematics, Article ID: 5498638, 1–9. Kilany, N. M. and H. M. Atallah (2018). Some characterizations of inverted beta and inverted beta lindley distributions. Journal Stat. Appl. Pro 7(3), 435–442. Kilany, N. M. and W. A. Hassanein (2018). Characterization of benktander typee II distribution via truncated moments and order statistics. International Journal of Probability and Statistics 7(4), 106–113. King, M. (2017). Statistics for Process Control Engineers: A Practical Approach (First ed.). New York, USA: Wiley. Korkmaz, M. C. (2019). A new family of the continuous distributions: the extended weibull-g family. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68(1), 248–270. Korkmaz, M. C. (2020). The unit generalized half normal distribution: a new bounded distribution with inference and application. U.P.B. Sci, Bull., Series A 82(2), 133–140. Korkmaz, M. C., C. Kus, and H. Erol (2011). The mixed weibull-negative binomial distribution. selcuk j. of applied mathematics. Special Issue, 21–33. Kuje, S. and K. E. Lasisi (2019). A transmuted lomax-exponential distribution: properties and applications. AJPAS 3(1), 1–13. Kumar, D. and M. Kumar (2019). A new generalization of the extended exponential distribution with an application. AODS 6(3), 441–462. Kumar, D., M. Nassar, A. Z. Afify, and S. Dey (2020). The complementary exponentiated lomax-poisson distribution with applications to bladder cancer and failure data. Austrian Journal of Statistics (forthcoming). Kumar, D., U. Singh, S. K. Singh, and P. K. Chaurasia (2018). A new lifetime distribution: some of its statistical properties and application. Journal Stat. Appl. Pro 7(3), 413–422. Lak, F., M. Basikhasteh, M. Alizadeh, and H. M. Yousof (2018). The odd log-logistic generalized half-normal lifetime poisson model. Pak.Journal stat.oper.res. 14(3), 111– 128. Lakshmi, M. V. and G. V. S. R. Anjaneyulu (2018). The odd generalized exponential type-i generalized half logistic distribution: properties and application. International Journal of Engineering and Computer Science 7(1), 23505–23516.

446

References

Lee, S. and J. H. T. Kim (2019). Exponentiated generalized pareto distribution: properties and applications towards extreme value theory. Communications in Statistics-Theory and Methods 48(8), 2014–2038. Lima, M. C. S. and G. M. Cordeiro (2017). The extended log-logistic distribution: properties and application. Annals of the Brazilian Academy of Sciences 89(1), 3–17. Lima, M. C. S., G. M. Cordeiro, and E. M. M. Ortega (2015). A new extension of the normal distribution. Journal of data Science 13(2), 385–408. Lima, M. C. S., G. M. Cordeiro, E. M. M. Ortega, and A. D. C. Nascimento (2019). A new extended normal regression model: simulations and applications. JSDA 6(7), 1–17. Lima, M. C. S., F. Prataviera, E. M. M. Ortega, and G. M. Cordeiro (2019). The odd loglogistic geometric family with applications in regression models with varying dispersion. (Forthcoming). Louzada, F., V. G. Cancho, and P. H. Ferreira (2020). The exponentiated poissonexponential distribution: a distribution with increasing, decreasing and bathtub failure rate. JSTA 19(2), 274–285. Louzada, F., I. Elbatal, and D. C. T. Granzotto (2018a). The beta exponentiated weibull geometric distribution: modeling, structural properties, estimation and an application to a cervical intraepithelial neoplasia dataset. Revista Brasileira De Biometria 36(4), 942– 967. Louzada, F., I. Elbatal, and D. C. T. Granzotto (2018b). The transmuted generalized lindley distribution: properties and an application to a data set on time-up-to-cure of patients treated with a triazole antifungal drug in an intensive care unit. Revista Brasileira De Biometria 36(2), 385–413. Madaki, U., M. Abu Bakar, and L. Handique (2018). Beta kumarswamy burr type x distribution and its properties. Mathematics and Computer Science Journal. Mahdavi, A. and L. Jabbari (2017). JMASM 16(1), 296–307.

An extended weighted exponential distribution.

Mahmoud, M. R. and R. M. Mandouh (2013). On the transmuted fréchet distribution. Journal of Applied Sciences Research 9(10), 5553–5561. Mahmoud, M, R., E. A. El-Sherpieny, and A. A. Ahmad (2015). The new kumaraswamy kumaraswamy family of generalized distributions with application. Pakistan Journal of Statistics and Operation Research 11, 159–180. Mahmoud, M, R., E. A. El-Sherpieny, and A. A. Ahmad (2016). The new kumaraswamy kumaraswamy weibull distribution with application. Pakistan Journal of Statistics and Operation Research 12, 165–184. Mahmoudi, E., R. Lalezari, and R. S. Meshkat (2017). A perturbed half-normal distribution and its applications. Journal of Statistical Research of Iran JSRI 14(2), 219–246.

References

447

Mahmoudi, E., H. Mahmoodian, and F. Esfandiari (2017). Normal-poisson distribution as a lifetime distribution of a series system. CIÊNCIAe NATURA 40(1-23). Mahmoudi, E., R. S. Meshkat, and M. Entezari (2020). A new class of beta-complementary exponential power series distributions. Journal of Testing and Evaluation 46(5), 2171– 2183. Mahmoudi, E., R. S. Meshkat, B. Kargar, and D. Kundu (2018). The extended exponentiated weibull distribution and its applications. Statistica 78(4), 363–396. Maiti, S. S. and S. Pramanik (2020). A generalized one parameter polynomial exponential generator family of distributions. arXiv:2006.05303v1. Makubate, B., M. Otlaadisa, B. O. Oluyede, A. F. Fagbamigbe, and A. Amey (2018). A new generalized class of linear failure rate power series distributions: model, theory and application. JDS 16(4), 801–828. Malik, A. S. and S. P. Ahmad (2019). Transmuted alpha power inverse rayleigh distribution: properties and application. Journal of Scientific Research 11(2), 185–194. Mallick, A. and I. Ghosh (2018). A new class of mixture probability models with applications. American Journal of Mathematical and Management Sciences 38(2), 207–226. Mandouh, R. M. and M. A. Mohamed (2020). Log-weighted pareto distribution and its statistical properties. JDS 18(1), 161–189. Mansoor, M., M. H. Tahir, A. Alzaatreh, and G. M. Cordeiro (2019). The poisson nadajahhaghighi distribution: properties and applications to lifetime data. International Journal of Reliability, Quality and Safety Engineering 27(1). Mansoor, M., M. H. Tahir, G. M. Cordeiro, S. Ali, and A. Alzaatreh (2020). The lindley negative-binomial distribution: properties estimation and applications to lifetime data. Mathematica Slovaca 70(4), 917–934. Mansour, M., E. Abd Elrazik, A. Afify, M. Ahsanullah, and E. Altun (2019). The transmuted transmuted-g family: properties and applications. Journal of Nonlinear Sciences and Applications 12, 217–229. Mansour, M., E. Abd Elrazik, E. Altun, A. Afify, and Z. Iqbal (2018). A new threeparameter fréchet distribution: properties and applications. Pak. Journal Statist 34(6), 441–458. Mansour, M., M. Rasekhi, M. Ibrahim, K. Aidi, H. M. Yousof, and E. A. Elrazik (2020). A new parameteric life distribution with modified bagdonaviˇcius-nikulin goodness-offit-test for censored validation, properties, applications and different estimation methods. Entropy 22(5), 592. Mansour, M. M., G. Aryal, A. Z. Afify, and M. Ahmad (2018). The kumaraswamy exponentiated fréchet distribution. Pak.Journal Statist. 34(3), 177–193.

448

References

Mansour, M. M., N. S. Butt, H. M. Yousof, S. I. Ansari, and M. Inrahim (2020). A generalization of reciprocal exponential model: Clayton copula, statistical properties and modeling skewed and symmetric real data sets. PJSOR 16(2), 373–386. Marinho, P. R. D., M. Bourguignon, R. B. Silva, and G. M. Cordeiro (2019). A new class of lifetime models and the evaluation of the confidence intervals by double percentile bootstrap. Anais de Academia Brasileira de Ciências 91(1), 1–27. Martinez-Florez, G., C. Barrera-Causil, and F. Marmolejo-Ramos (2020). The exponentialcentred skew-normal distribution. Symmetry (MDPI) 12(7), 1140. Marzouk, W., L. Handique, A. H. N. Ahmed, F. Jamal, and A. A. A. Rahman (2020). The generalized odd linear exponential family of distributions with applications to reliability theory. (Forthcoming). Marzouk, W., F. Jamal, A. H. N. Ahmad, and A. A. E. Ahmed (2019). The generalized odd lomax generated family of distribution with applications. Gazi University Journal of Science 32(2), 737–755. Mashabe, B., B. O. Oluyede, A. F. Fagbamigbe, B. Makubate, and S. Krishnarani (2019). A new class of generalized weibull-g family of distributions: theory, properties and applications. IJSP 8(1), 73–93. Maurya, S., S. K. Singh, and U. Singh (2020). A new distribution with monotone and non-monotone shaped failure rate. Journal of Applied Probability and Statistics 15(1), 1–21. Maxwell, O., A. U. Chukwu, O. S. Oyamakin, and M. A. Khaleel (2019). The marshallolkin inverse lomax distribution (mo-ild) with application to cancer stem cell. Journal of Advances in Mathematics and Computer Science 33(4), 1–12. Maxwell, O., A. A. Kayode, I. P. Onyedikachi, C. I. Obi-Okpala, and E. U. Victor (2019). Useful generalization of the inverse lomax distribution: statistical properties and application to lifetime data. American Journal of Biomedical Science & Research 6(3), 258–265. Maxwell, O., O. S. Oyamakin, and J. T. Eghwerido (2019). The gompertz length biased exponential distribution and its application to uncensored data. Current Trends on Biostatistics and Biometrics 1(3), 52–57. Maxwell, O., S. O. Oyamakin, A. U. Chukwu, Y. O. Olusola, and A. A. Kayode (2019). New generalization of length biased exponential distribution with applications. Journal of Advances in Applied Mathematics 4(2), 82–88. Mazucheli, J., S. R. Bapat, and A. F. B. Menezes (2020). A new one-parameter unit-lindley distribution. Chilean Journal of Statistics 11(1), 53–67. Mazucheli, J., A. F. B. Menezes, and S. Chakraborty (2019). On the one parameter unitlindley distribution and its associated regression model for proportion data. Journal of Applied Statistics 46(4), 700–714.

References

449

Mazucheli, J., A. F. B. Menezes, and S. Dey (2018). The unit-birnbaum-saunders distribution with applications. Chilean Journal of Statistics 19(1), 47–57. Mazucheli, J., A. F. B. Menezes, and M. E. Ghitany (2018). The unit-weibull distribution and associated inference. Journal of Applied Probability and Statistics 13(2), 1–22. Mdlongwa, P., B. O. Oluyede, A. Amey, A. F. Fagbamigbe, and B. Makubate (2019). Kumaraswamy log-logistic weibull distribution: model, theory and application to lifetime and survival data. Heliyon 5(1). Mdlongwa, P., B. O. Oluyede, A. Amey, and S. Huang (2018). The burr XII modified weibull distribution: model, properties and applications. Electronic Journal of Applied Statistical Analysis 10(1), 118–145. Mead, M., A. Afify, and N. Butt (2020). The modified kumarswamy weibull distribution: properties and applications in reliability and engineering sciences. PJSOR 16(3), 433– 446. Mead, M. E. (2015). The kumaraswamy exponentiated burr XII distribution and its applications. Pak.Journal stat.oper.res. 11(2), 138–148. MirMostafaee, S. M. T. K., M. Alizadeh, E. Altun, and S. Nadarajah (2019). The exponentiated generalized power lindley distribution: properties and applications. Appl. Math. Journal Chinese Univ 34(2), 127–148. Mirzadeh, S. and A. Iranmanesh (2019). A new class of skew-logistic distribution. Mathematical Sciences 13(375-385), 306–8. Mirzaei, S., G. Mohtashami Borzadaran, M. Amin, and H. Jabbari (2019). A new generalized weibull distribution in income economic inequality curves. Communications in Statistics-Theory and Methods 48(4), 889–908. Modi, K., D. Kumar, and Y. Singh (2020). A new family of distributions with application on two real datasets. on survival problem. Science & Technology Asia 25(1), 1–10. Mohammed, A. S. and F. I. Ugwuowo (2020). A new family of distributions for generating skewed models: properties and applications. Pak. Journal Statist. 36(2), 149–168. Momenkhan, F. A. (2019). Transmuted extended lomax distribution with some tractability properties and applications. International Journal of Advanced Statistics and Probability (JASP) 7(1), 28–33. Mozafari, M., M. Afshari, M. Alizadeh, and H. Karamikabir (2019). The zografosbalakrishnan odd log-logistic generalized half-normal distribution with mathematical properties and simulations. Statistics, Optimization and Information Computing 7, 211– 234. Mudiasir, S. and S. P. Ahmad (2017). Weighted nakagami distribution with applications using r-software. In International conference on recent innovations in science, agriculture, engineering and management, pp. 797–809.

450

References

Mudiasir, S. and S. P. Ahmad (2018). Weighted version of generalized inverse weibull distribution. Journal of Modern Applied Statistical Methods 17(2), 1–22. Muhammad, M. (2020). Poisson-odd generalized exponential family of distributions: theory and applications. (Preprint). Muhammad, M., I. Muhammad, and A. M. Yaya (2018). The kumaraswamy exponentiated u-quadratic distribution: properties and application. Asian Journal of Probability and Statistics (AJPAS) 1(3), 1–17. Muhammad, M. and M. I. Suleiman (2019). The transmuted exponentiated u-quadratic distribution for lifetime modeling. Sohag Journal Math. (forthcoming). Mukhtar, S., A. Ali, and A. M. Alya (2018). Mcdonald modified burr-III distribution: properties and applications. Journal of Taibah University for Science 13(1), 184–192. Nadarajah, S., V. Nassiri, and A. Mohammadpour (2009). Truncated-exponential skewsymmetric distributions. (Personal Communication). Nasir, M. A., F. Jamal, and C. Chesneau (2020). The odd generalized gamma-g family of distributions: properties, regressions and applications. Statistica 80(1), 3–38. Nasir, M. A., F. Jamal, and A. A. Shah (2019). A new compound four-parameter lifetime model: properties, cure rate model and applications. HAL Id: hal-01902847; hal.archives-ouvertes.fr/hal-01902847v1, 1–22. Nasir, M. A., F. Jamal, G. O. Silva, and M. H. Tahir (2018). Odd burr-g poisson family of distributions. Journal of Statistics Applications and Probability 7(1), 9–28. Nasir, M. A., M. C. Korkmaz, F. Jamal, and H. M. Yousof (2018). On a new weibull burr XII distribution for lifetime data. Sohag Journal of Mathematics 5(2), 47–56. Nasir, M. A., G. Ozel, and F. Jamal (2018). The burr XII uniform distribution: theory and applications. Journal of Reliability and Statistical Studies 11(2), 143–158. Nasir, M. A., M. H. Tahir, C. Chesneau, F. Jamal, and M. A. A. Shah (2020). The odds generalized gamma-g family of distributions: properties, regressions and applications. Statistica 80(1), 3–38. Nasir, M. A., H. M. Yousof, F. Jamal, and M. C. Korkmaz (2019). The exponentiated burr XII power series distributions: properties and applications. Stats 2(1), 15–31. Nasiru, S. (2018). Extended odd fréchet-g family of distributions. Journal of Probability and Statistics, Article ID: 2931326, 1–12. Nasiru, S., A. G. Abubakari, and J. Abonongo (2020). Unit nadarajah-haghighi generated family of distributions: properties and applications. Sankhya A: The Indian Journal of Statistics, 1–27.

References

451

Nasiru, S., B. A. M. Atem, and K. Nantomah (2018). Poisson exponentiated erlangtruncated exponential distribution. Journal of Statistics Applications and probability 7(2), 245–261. Nasiru, S., P. N. Nwita, and O. Ngesa (2019). Exponentiated generalized power series family of distributions. AODS 6(3), 463–489. Nassar, M., A. Z. Afify, S. Dey, and D. Kumar (2018). A new extension of weibull distribution: properties and different methods of estimation. Journal of Computational and Applied Mathematics 336, 439–457. Nassar, M., A. Alzaatreh, M. Mead, and O. Abo-Kasem (2015). Alpha power weibull distribution: Properties and applications, commun. Communications in Statistics-Theory and Methods 46(20), 10236–10252. Nassar, M., S. Dey, and D. Kumar (2018). Logarithm transformed lomax distribution with applications. Calcutta Statistical Association 70(2), 122–135. Nassar, M., D. Kumar, S. Dey, G. M. Cordeiro, and A. Z. Afify (2019). The marshall-olkin alpha power family of distributions with applications. Journal of Computational and Applied Mathematics 351, 41–53. Nawaz, S., M. H. Tahir, and A. Javaid (2020). On some extension of beta power function distribution. Asian Journal of Mathematics and Statistics 13(1), 1–6. Nofal, Z. M., A. Z. Afify, H. M. Yousof, and G. M. Cordeiro (2019). The generalized transmuted-g family of distributions. Communications in Statistics-Theory and Methods 46(8), 4119–4136. Nofal, Z. M., A. Z. Afify, H. M. Yousof, D. C. T. Granzotto, and F. Louzada (2018). The transmuted exponentiated additive weibull distribution: properties and applications. JMASM 17(1), 2–32. Nofal, Z. M. and M. Ahsanullah (2018). A new extension of the fréchet distribution: properties and characterization. Communications in Statistics-Theory and Methods 48(9), 2267–2285. Odom, C. C. and M. A. Ijomah (2019). Odoma distribution and its application. Asian Journal of Probability and Statistics (AJPAS) 4(1), 1–11. Ogunde, A. A., F. Olayode, and A. Adejumoke (2020). Cubic transmuted gompertz distribution: as a life time distribution. Journal of Advances in Mathematics and Computer Science 35(1), 105–116. Oguntunde, P. E. (2015). On the exponentiated weighted exponential distribution and its basic statistical properties. Applied Science Reports 10(3), 160–167. Oguntunde, P. E. and A. O. Adejumo (2015). The generalized inverted generalized exponential distribution with an application to a censored data, j. Journal of Statistics Applications & Probability 4(2), 223–230.

452

References

Oguntunde, P. E., K. A. Ilori, and H. Okagbue (2018). The inverted weighted exponential distribution with applications. International Journal of Advances and Applied Sciences 5(11), 46–50. Oguntunde, P. E., M. A. Khaleel, M. T. Ahmed, A. O. Adejumo, and O. A. Odetunmibi (2017). A new generalization of the lomax distribution with increasing and constant failure rate. Modeling and Simulation in Engineering. Oguntunde, P. E., M. A. Khaleel, M. T. Ahmed, and H. I. Okagbue (2020). The gompertz fréchet distribution: properties and applications. Cogent Mathematics & Statistics 6(1), 1568662. Oguntunde, P. E., M. A. Khaleel, H. Okagbue, and O. A. Odetunmibi (2019). The toppleone lomax (tllo) distribution with application to airborne communication yransceiver dataset. Wireless Personal Communications 109(1), 349–360. Oguntunde, P. E., E. A. Owoloko, and O. S. Balogun (2016). On a new weighted exponential distribution: theory and application. Asian Journal of Applied Sciences 9(1), 1–12. Okoli, O. C., G. A. Osuji, D. F. Nwosu, and K. N. C. Njoku (2016). On the modified extended generalized exponential distribution. European Journal of Statistics and Probability (EJSP) 4(4), 1–11. Okorie, I. E., A. C. Akpanta, and J. Ohakwe (2017). The marshall-olkin extended power function distribution. European Journalof Statistics and Probability 5(3), 16–29. Okorie, I. E., A. C. Akpanta, J. Ohakwe, and D. C. Chikezie (2017). The extended erlangtruncated exponential distribution and application to rainfall data. Heliyon, Mathematics 3(6), 1–39. Okorie, I. E., A. C. Akpanta, J. Ohakwe, D. C. Chikezie, and C. U. Onyemachi (2019). On the rayleigh-geometric distribution with application. Heliyon 5, 1–10. Okorie, I. E., A. C. Akpanta, J. Ohakwe, D. C. Chikezie, C. U. Onyemachi, and M. K. Rastogi (2020). Zero-truncated poisson-power function distribution. Annals of Data Science, 1–23. Olayode, F. (2019). The topp-leone rayleigh distribution with application. American Journal of Mathematics and Statistics 9(6), 215–220. Olmos, N. M., O. Venegas, Y. M. Gomez, and Y. A. Iriarte (2019). An asymmetric distribution with heavy tails and its expectation-maximization (em) algorithm implementation. Symmetry (MDPI) 11(9), 1–13. Oluyede, B. O., B. Makubate, and A. F. Fagbamigbe (2019). A new burr XII-weibulllogarithmic (bwl) distribution for survival and lifetime data analysis: model, theory and applications. STATS 1(1), 77–91. Oluyede, B. O., P. Mdlongwa, B. Makubate, and S. Huang (2019). The burr-weibull power series class of distributions. Austrian Journal of Statistics 48(1), 1–13.

References

453

Opone, F. C., N. Ekhosuehi, and S. E. Omosigho (2020). Topp-leone power lindley distribution (tlpld): its properties and application. Sankhya A: The Indian Journal of Statistics; 12. Ortega, E. M. M., J. N. da Cruz, and G. M. Cordeiro (2019). The log-odd logistic-weibull regression model under informative censoring. Model Assisted Statistics and Applications 14(3), 239–254. Ortega, E. M. M., A. J. Lemonte, G. M. Cordeiro, and J. N. da Cruz (2016). The odd birnbaum-saunders regression model with applications to lifetime data. Journal of Statistical Theory and Practice 10(4), 780–804. Osatohanmwen, P., F. O. Oyegue, F. Ewere, and B. Ajibade (2020). A new family of generalized distributions on the unit interval: the t-kumaraswamy family of distributions. JDS 18(2), 218–237. Osatohanmwen, P., F. O. Oyegue, and S. M. Ogbonmwan (2019). A new member of the t-x family of distributions: the gumbel-burr XII distribution and its properties. Sankhya A 81(2), 298–322. Osatohanmwen, P., F. O. Oyegue, and S. M. Ogbonmwan (2020). The T-R {Y} power series family of probability distributions. Journal of the Egyptian Mathematical Society 28(1), 1–18. Otiniano, C. E. G., B. S. de Paiva, and D. S. B. M. Neto (2019). The transmuted gev distribution: properties and application. Communications for Statistical Applications and Methods 26(3), 239–259. Ozel, G., E. Altun, M. Alizadeh, and M. Mozafzri (2018). The odd log-logistic log-normal distribution with theory and applications. Advances in Data Science and Adaptive Analysis 10(04), 1850009. Paranaíba, P. F., G. M. Cordeiro, and A. K. Ortega, E. M. M.and Suzuki (2019). The odd log-logistic geometric normal regression model with applications. Advances in Data Science and Adaptive Analysis 11(01n02), 1950003. Paranaíba, P. F., E. M. M. Ortega, G. M. Cordeiro, and M. de Pascoa (2013). The kumaraswamy burr XII distribution:theory and practice. Journal of Statistical Computation and Simulation 83(11), 2117–2143. Pena-Ramirez, F. A., R. R. Guerra, and G. M. Cordeiro (2019). The nadarajah-haghighi lindley distribution. Anais de Academia Brasileira de Ciências 91(1), 1. Péna-Ramírez, F. A., R. R. Guerra, G. M. Cordeiro, and P. R. D. Marinho (2018). The exponentiated power generalized weibull: properties and applications. Anais Da Academia Brasileira de Ciencias 90(3), 2553–2577. Pillai, J. K. and G. B. Moolath (2019). A new generalization of the fréchet distribution: properties and application. Statistica 79(3), 267–289.

454

References

Pinho, L. G. B., G. M. Cordeiro, and J. S. Nobre (2015). On the harris-g class of distributions: general results and application. Brazilian Journal of Probability and Statistics 29(4), 813–832. Pobo´ciková, I., Z. Sedliaˇcková, and M. Michalková (2018). Transmuted weibull distribution and its applications. MATEC Web of Conferences 157, 1–11. Pourdarvish, A., S. Mir Mostafaee, and K. Naderi (2015). The exponentiated topp-leone distribution: properties and application. Journal of Applied Environmental and Biological Sciences 5(7), 251–256. Prataviera, F., G. M. Cordeiro, E. M. M. Ortega, and A. K. Suzuki (2019). The odd loglogistic geometric normal regression model with applications. Advances in Data Science and Adaptive Analysis 11(01n02), 1950003. Qoshja, A. and M. Muça (2018). A new modified generalized odd log-logistic distribution with three parameters. Mathematical Theory and Modeling 8(1), 49–60. Rahman, M., B. Al-Zahrani, and M. Q. Shahbaz (2018a). A general transmuted family of distributions. Pak.Journal stat.oper.res. 14(2), 451–469. Rahman, M., B. Al-Zahrani, and M. Q. Shahbaz (2018b). New general transmuted family of distributions with applications. Pak.Journal stat.oper.res. 14(4), 807–829. Rahman, M., B. Al-Zahrani, and M. Q. Shahbaz (2019). Cubic transmuted weibull distribution: properties and applications. Annals of Data Science (AODS) 5(1), 83–102. Rahman, M., B. Al-Zahrani, and M. Q. Shahbaz (2020). Cubic transmuted pareto distribution. AODS 7(1), 91–108. Ramires, T. G., G. M. Cordeiro, M. W. Kattan, N. Hens, and E. M. M. Ortega (2017). Predicting the cure rate of breast cancer using a new regression model with four regression structures. Statistical Methods in Medical Research 27(11), 3207–3223. Ramires, T. G., E. M. M. Ortega, M. Cordeiro G., and N. Hens (2018). A flexible bimodal with long-term survivors and different regression structures. Communications in Statistics-Simulation and Computation, 1–22. Ramos, P. L. and F. Louzada (2019). A distribution for instantaneous failures. STATS 2, 247–258. Ramos, P. L., F. Louzada, T. K. Shimizu, and A. O. Luiz (2019). The inverse weighted lindley distribution: properties, estimation and an application on a failure time data. Communications in Statistics-Theory and Methods 48(10), 2372–2389. Ranjbar, V., M. Alizadeh, and E. Altun (2019). Extended generalized lindley distribution: properties and applications. Journal of Mathematical Extension 13(1), 117–142. Rao, G. S. and S. Mbwambo (2019). Exponentiated inverse rayleigh distribution and an application to coating weights of iron sheets data. Journal of Probability and Statistics, Article ID: 7519429 2019, 1–13.

References

455

Rasekhi, M., M. Alizadeh, and G. G. Hamedani (2018). The kumarswamy weibull geometric distribution with applications. PJSOR 14(2), 359–378. Rasheed, N. (2019). Topp-leone compound rayleigh distribution: properties and applications. Research Journal of Mathematics and Statistical Sciences 7(3), 51–58. Rather, A. A. and G. Ozel (2020). The weighted power lindley distribution with applications on the life time data. PJSOR 16(2), 225–237. Rather, A. A. and C. Subramanian (2018). Exponentiated mukherjee-islam distribution. Journal Stat. Appl. Pro 7(2), 357–361. Rather, A. A. and C. Subramanian (2020). A new exponentiated distribution with engineering science applications. Journal Stat. Appl. Pro 9(1), 127–137. Refaie, M. K. A. (2018a). Extended poisson-exponentiated weibull distribution: theoretical and computational aspects. Pak. Journal Statist. 34(6), 513–530. Refaie, M. K. A. (2018b). A new extension of the burr type XII distribution. Journal of Mathematics and Statistics 14, 261–274. Refaie, M. K. A. (2019). A new two-parameter exponentiated weibull model with properties and applications to failure and survival times. International Journal of Mathematical Archive 10(2), 1–13. Reyad, H., A. Z. Afify, F. Jamal, and S. Othman (2019). The extended power lindley-g family of distributions: properties and applications. Journal of Modern Applied Statistical Methods (JMASM). (Forthcoming). Reyad, H., M. Alizadeh, F. Jamal, and S. Othman (2018). The topp leone odd lindley-g family of distributions: properties and applications. Journal of Statistics & Management Systems 21(7), 1273–1297. Reyad, H., F. Jamal, S. Othman, and N. Yahia (2019). The topp leone generalized inverted kumarswamy distribution: properties and applications. Asian Research Journal of Mathematics 13(3), 1–15. Reyad, H., F. Jamal, G. Ozel, and S. Othman (2020). The poisson exponential-g family of distributions with properties and applications. Journal of Statistics & Management System, 1–24. Reyad, H., S. Othman, and M. Ahsan ul Haq (2019, Journal Data Science). The transmuted generalized odd generalized exponential-g family of distributions: theory and applications. Journal of Data science 17(2), 279–298. Reyad, H., S. M. and S. Othman (2019). The nadarajah haghighi topp leone-g family of distributions with mathematical properties and applications. Pakistan Journal of Statistics and Operation Research 4, 849–866. Reyes, J., J. Iriarte, Y. A., and H. Gómez (2018). The slash lindley-weibull distribution. Methodology and Computing in Applied Probability 21(1), 235–251.

456

References

Rezaei, S, M., S. A. K., Nadarajah, and M. Alizadeh (2016). A new exponentiated class of distributions: properties and applications. Communications in Statistics-Theory and Methods 46(12), 6054–6073. Rezk, H. (2020). Extended reciprocal rayleigh distribution: Copula, properties and real data modeling. Pakistan Journal of Statistics and Operation Research 16(1), 35–52. Riffi, M. (2018). A generalized transmuted gompertz-makeham distribution. Journal of Scientific and Engineering Research 5(8), 252–266. Riffi, M. I., S. I. Ansari, and M. Hamdan (2019). A generalized transmuted fréchet distribution. Journal Stat. Appl. Pro 8(2), 1–10. Rodrigues, J. d. A. and A. P. C. M. Silva (2015). The exponentiated kumaraswamyexponential distribution. British Journal of Applied Science & Technology 10(5), 1–12. Saboor, A., I. Elbatal, M. N. Khan, G. M. Cordeiro, and R. R. Pescim (2018). The beta exponentiated nadarajah-haghighi distribution: theory and applications. HAL Id: hal01570564, 1–17. Saboor, A., M. N. Khan, G. M. Cordeiro, M. A. R. Pascoa, J. Bortolini, and S. Mubeen (2018). Modified beta modified-weibull distribution. Computational Statistics 34(1), 173–199. Saboor, A., M. N. Khan, G. M. Cordeiro, M. A. R. Pascoa, P. L. Ramos, and M. Kamal (2019). Some new results for the transmuted generalized gamma distribution. Journal of Computational and Applied Mathematics 352, 165–180. Samuel, A. F. and O. A. Kehinde (2019). A study of transmuted half logistic distribution: properties and application. International Journal of Statistical Distributions and Applications 5(3), 54–59. Sangsanit, Y. and W. Bodhisuwan (2016). The topp-leone generator of distributions: properties and inferences,. Songklanakarin Journal of Science & Technology 38(5), 537–548. Sanusi, A. A., S. I. S. Doguwa, I. Audu, and Y. M. Baraya (2020). Burr x exponential-g family of distributions: properties and application. AJPAS 7(3), 58–75. Sanusi, A. A., S. I. S. Doguwa, A. I. Ishaq, T. Musa, and A. Usman (2019). Burr x-kumaraswamy distribution: properties and application. Researchgate.net/publication/334576668. Saraço˘glu, B. and C. Tani¸s (2018). A new statistical distribution: cubic transmuted kumaraswamy distribution and its properties. Journal of the National Science Foundation of Sri Lanka 46(4), 505–518. Sarma, S. and K. K. Das (2020). Weighted inverse nakagami distribution. International Journal of Mathematical and Computational Methods 5, 14–25. Sebastian, N., R. S. Rasin, and P. O. Silviya (2019). Topp-leone generated q-exponential distribution and its applications.

References

457

Segovia, F. A., Y. M. Gomez, O. Venegas, and H. W. Gomez (2020). A power maxwell distribution with heavy tails and applications. Mathematics (MDPI) 8(7), 1–20. Sen, S., A. Z. Afify, H. Al-Mofleh, and M. Ahsanullah (2019). The quasi xgammageometric distribution with application in medicine. Filomat 33(16), 5291–5330. Sen, S. and N. Chandra (2016). The leaned normal distribution: an alternative to skewed data analysis. ResearchGate.net/publication/307578175, 11 pages. Sen, S., N. Chandra, and S. S. Maiti (2017). The weighted xgamma distribution: properties and application. Journal of Reliability and Statistical Studies 10(1), 43–58. Sen, S., M. C. Korkmaz, and H. M. Yousof (2018). The quasi xgamma-poisson distribution: properties and application. Journal of the Turkish Statistical Association 11(3), 65–76. Sen, S., S. S. Maiti, and N. Chandra (2016). The xgamma distribution: statistical properties and application. Journal of Modern Applied Statistical Methods (JMASM) 15(1), 774– 788. Shah, S., S. Chakraborty, and P. J. Hazarika (2019). A new one parameter bimodal skew logistic distribution and its applications. Preprint, 17 pages. Shah, S., S. Chakraborty, P. J. Hazarika, and M. M. Ali (2020). The log-balakrishnanalpha-skew-normal distribution and its applications. Pakistan Journal of Statistics and Operation Research 16(1), 109–117. Shah, S. and P. J. Hazarika (2019). The alpha beta skew logistic distribution and its properties. Technical report, Preprint, 15 pages. Shahid, N., R. Khalil, and J. Khokhar (2020). Zografos balakrishnan power lindley distribution. Journal of Data Science 18(2), 279–298. Shahzad, M. N., F. Merovci, and G. Asghar (2017). Transmuted singh-maddala distribution: a new flexible and upside-down bathtub shaped hazard function distribution. Revista Colombiana de Estadistica 40(1), 1–27. Shakhatreh, M. K., A. J. Lemonte, and G. M. Cordeiro (2019). On the generalized extended exponential-weibull distribution: properties and different methods of estimation. International Journal of Computer Mathematics 97(5), 1029–1057. Shanker, R. (2015). Akash distribution and its applications. International Journal of Probability and Statistics 4(3), 65–75. Shanker, R. (2016). A quasi sujatha distribution. International Journal of Probability and Statistics 5(4), 89–100. Shanker, R., K. K. Shukla, and H. Fesshaye (2017). A generalization of sujatha distribution and its applications with real lifetime data. Journal of Institute of Science and Technology (JIST) 22(1), 66–83.

458

References

Shanker, R., K. K. Shukla, and T. A. Leonida (2019). A generalization of two-parameter lindley distribution with properties and applications. Journal of Probability and Statistics 8(1), 1–13. Sharma, V. K., M. M. Ali, M. C. Korkmaz, H. M. Yousof, and M. Ibrahim (2020). A generalization of lindley distribution: copula. properties and different methods of estimation. (Personal Communication). Shawki, A. W. and M. Elgarhy (2017). Kumarswamy sushila distribution. International Journal of Scientific Engineering and Science 1(7), 29–32. Shekari, M., H. Zamani, and M. M. Saber (2019). The compound class of janardan-power series distribution: properties and applications. Journal of Data Science 17(2), 259–278. Shrahili, M., I. Elbatal, and M. Muhammad (2019). The type i half-logistic burr x distribution: theory and practice. Journal of Nonlinear Science and Applications 12(5), 262–277. Silva, G. O., G. M. Cordeiro, and E. M. M. Ortega (2020). Surviving and non surviving fraction regression models based on the beta modified weibull distribution. Model Assisted Statistics and Applications 15(2), 111–126. Silva, R., F. Gomes-Silva, M. Ramos, G. Cordeiro, P. Marinho, and T. A. N. De Andrade (2019). The exponentiated kumaraswamy-g class: general properties and application. Revista Colombiana de Estadistica 42(1), 1–33. Silveira, F. V. J., F. Gomes-Silva, C. C. R. Brito, M. Cunha-Filho, F. R. S. Gusmao, and S. F. A. Xavier-Junior (2019). Normal-c class of probability distributions: properties and applications. Symmetry 11, 1–17. Song, P. and S. Wang (2019). Generalized log-lindley distribution and its applications in stochastic comparison. Communications in Statistics-Theory and Methods 48(15), 4008– 4018. Tablada, C. J. and G. M. Cordeiro (2017). The modified fréchet distribution and its properties. Communications in Statistics-Theory and Methods 46(21), 10617–10639. Tahir, M. H., G. M. Cordeiro, S. Ali, S. Dey, and A. Manzoor (2018). The inverted nadarajah-haghighi distribution: estimation methods and applications. Journal of Statistical Computation and Simulation 88(14), 2775–2798. Tahir, M. H., G. M. Cordeiro, A. Alzaatreh, M. Mansoor, and M. Zubair (2015). A new weibull-pareto distribution: properties and applications. Communications in StatisticsSimulation and Computation 45(10), 3548–3567. Tahir, M. H., M. Zubair, G. M. Cordeiro, A. Alzaatreh, and M. Mansoor (2016). The poisson-x family of distributions. Journal of Statistical Computation and Simulation 86(14), 2901–2921.

References

459

Tahir, M. H., M. Zubair, M. Mansoor, G. M. Cordeiro, M. Alizadeh, and G. G. Hamedani (2016). A new weibull-g family of distributions. Hacettepe Journal of Mathematics and Statistics 45(2), 629–647. Tanis, C., B. Saracoglu, C. Kus, and A. Pekgor (2020). Transmuted complementary exponential power distribution: properties and applications. Cumhuriyet Science Journal 41(2), 419–432. Tarvirdizade, B. and N. Nematollahi (2019a). A new flexible hazard rate distribution: application and estimation of its parameters. Communications in Statistics-Simulation and Computation 48(3), 882–899. Tarvirdizade, B. and N. Nematollahi (2019b). The power-linear hazard rate distribution and estimation of its parameters under progressively type-II censoring. Hacettepe Journal of Mathematics and Statistics 48(3), 818–844. Teamah, A. A. M., A. A. Elbanna, and A. M. Gemeay (2020). Fréchet-weibull distribution with applications to earthquakes data set. Pakistan Journal of Statistics 36(2), 135–147. Tesfay, M. and R. Shanker (2018a). A new two-parameter sujatha distribution with properties and applications. Turkiye Klinikleri Journal of Biostatistics 10(2), 96–113. Tesfay, M. and R. Shanker (2018b). A two-parameter sujatha distribution. Biometrics & Biostatistics International Journal 7(3), 188–197. Tesfay, M. and R. Shanker (2019). Another two-parameter sujatha distribution with properties and applications. Journal of Mathematical Sciences and Modelling 2(1), 1–13. Tharshan, R. and P. Wijekoon (2020). Location based generalized akash distribution: properties and applications. Open Journal of Statistics 10(2), 163–187. Theodossiou, P. (2019). Skewed type III generalized logistic distribution. Communications in Statistics-Theory and Methods 48(23), 5809–5819. Thm, A. (2018). A new flexible distribution based on the zero truncated poisson distribution: mathematical properties and applications to lifetime data. Biostatistics and Biometrics Journal (JP Juniper) 8(1), 1–7. Udoudo, U. P. and E. H. Etuk (2018). A new extension of quasi lindley distribution: properties and applications. International Journal of Advanced Statistics and Probability 7(2), 28–41. Umar, M. A. and W. B. Yahya (2020). A new exponential-gamma distribution with applications. Journal of Modern Applied Statistical Methods. Umar, N., D. A. Saddam, and Y. Abdulkadir (2019). Odd burr III exponential distribution: its theory and application. (ResearchGate). Ünal, C., S. Cakymakyapan, and G. Ozel (2018). Alpha power inverted exponential distribution: properties and application. Gazi University Journal of Science 31(3), 954–965.

460

References

Usman, R. M., B. N. Ahsan ul Haq, M., and G. Özel (2018). Exponentiated transmuted power function distribution: theory & applications. Gazi University Journal of Science 31(2), 660–675. Usman, R. M. and M. Ahsan ul Haq (2019). Some remarks on odd burr III weibull distribution. Annals of Data Science 6(1), 21–38. Usman, U., S. Shamsuddeen, B. M. Arkilla, and Y. Aliyu (2020). Inferences on the weibull exponentiated exponential distribution and applications. International Journal of Statistical Distributions and Applications 6(1), 10–22. Uwadi, U. U. and E. E. Nwaezza (2018). A new generalized transmuted inverse exponential distribution: properties and application. Asian Journal of Probability and Statistics 2(1), 1–13. Uwadi, U. U., E. W. Okereke, and C. O. Omekara (2019). Exponentiated gumbel exponential distribution: properties and applications. American Journal of Applied Mathematics and Statistics 7(5), 178–186. Uwaeme, O. R., N. P. Akpan, and U. C. Orumie (2018). An extended pranav distribution. Asian journal of Probability and Statistics 2(4), 1–15. Vasconcelos, J. C. S., G. M. Cordeiro, E. M. M. Ortega, and E. G. Araújo (2019). The new odd log-logistic generalized inverse Gaussian regression model. Journal of Probability and Statistics, Article ID: 8575424, 1–13. Vasconcelos, J. C. S., G. M. Cordeiro, E. M. M. Ortega, and M. A. M. Biaggioni (2020). The parameteric and additive partial linear regressions based on the generalized odd loglogistic log-normal distribution. Communications in Statistics-Theory and Methods, 1– 28. Vasconcelos, J. C. S., G. M. Cordeiro, E. M. M. Ortega, and E. M. de Rezende (2020). The new regression model for bimodal data and applications in agriculture. Journal of Applied Statistics (forthcoming). Yadav, A. S., E. Altun, and H. M. Yousof (2020). Burr-hatke exponential distribution: a decreasing failure rate model, statistical inference and applications. Annals of Data Science, 1–20. Yadav, A. S., H. Goual, R. M. Alotaibi, H. Rezk, M. M. Ali, and H. M. Yousof (2020). Validation of the topp-leone-lomax model via a modified nikulin-rao-tobson goodnessof-fit test with different methods of estimation. Symmetry (MDPI) 12(57), 1–29. Yadav, A. S., S. S. Maiti, and M. Saha (2020). The inverse xgamma distribution: statistical properties and different methods of estimation. Annals of Data Science, 1–19. Yarmoghaddam, N. and E. B. Samani (2019). A new lifetime distribution: the beta modified weibull power series distribution. Journal of Statistical Research of Iran 16(1), 1–31.

References

461

Yazar, O. and M. C. Korkmaz (2016). A generalization of the exponential and lindley distributions via the kumarswamy-g family. 2nd International Conference on Analysisi and its Applications, 12–15. Yousof, H. M., M. Ahsanullah, and M. G. Khalil (2019). A new zero truncated version of the poisson burr XII distribution: characterizations and properties. Journal of Statistical Theory and Applications 18(1), 1–11. Yousof, H. M., E. Altun, T. G. Ramires, M. Alizadeh, and M. Rasekhi (2018). A new family of distributions with properties, regression models and applications. Journal of Statistics and Management Systems 21(1), 163–188. Yousof, H. M., M. C. Korkmaz, and S. Sen (2020). A new two-parameter lifetime model. Annals of Data Science, 1–16. Yousof, H. M., M. Majumder, S. M. A. Jahanshahi, A. Massom, and G. G. Hamedani (2019). A new weibull class of distributions. (To appear in PJS). Yousof, H. M., M. Rasekhi, E. Altun, and M. Alizadeh (2019). The extended odd fréchet family of distributions: properties, applications and regression modeling. International Journal of Applied Mathematics and Statistics 30(1), 1–30. Yusuf, A. and S. Qureshi (2019). A five parameter statistical distribution with application to real data. Journal Statistics Applications & Probability 8(1), 11–26. Zaka, A., A. S. Akhtar, and R. Jabeen (2020). The new reflected power function distribution: theory, simulation & application. AIMS Mathematics 5(5), 5031–5054. Zamani, Z., M. Afshari, H. Karamikabir, M. Khan Ahmadi, and M. Alizadeh (2020). The new extension of odd log-logistic chen distribution: mathematical properties and applications. Thailand Statistician (forthcoming). ZeinEldin, R. A. (2020). Ahsan ul haq, m., hashmi, s. and elseherty, m. (2020). alpha power transformed inverse lomax distribution with different methods of estimation and applications. Complexity Article ID:1860813, 1–15. ZeinEldin, R. A. and M. Elgarhy (2018). A new generalization of weibull-exponential distribution with application. Journal of Nonlinear Science and Applications 11, 1099– 1112. ZeinEldin, R. A., F. Jamal, C. Chesneau, and M. Elgarhy (2019). Type II topp-leone inverted kumarswamy distribution with statistical inference and applications. Symmetry 11(12), 1459. Zelibe, S. C., E. J. Thomas, and E. Efe-Eyefia (2019). Kumaraswamy alpha power inverted exponential distribution: properties and applications. Istatistik Journal of the Turkish Statistical Association 12(2), 35–48. Zelibe, S. C., E. J. Thomas, E. Efe-Eyefia, and E. Ekuma-Okereke (2019). On the extended new generalized exponential distribution: properties and applications. FUPRE Journal of Scientific and Industrial Research 3(1), 112–122.

462

References

Zichuan, M., S. Hussain, A. Iftikhar, M. Ilyas, Z. Ahmad, D. M. Khan, and S. Mansoor (2020). A new extended-x family of distributions: properties and applications. Computational and Mathematical Methods in Medicine, Article ID: 46505209, 1–13. Zubair, M., A. Alzaatreh, M. H. Tahir, M. Mansoor, and M. Mustafa (2018). A generalization of the exponential distribution and its applications on modelling skewed data. Statistical Theory and Related Fields 2(1), 68–79. Zubair, M., T. K. Pogany, G. M. Cordeiro, and M. H. Tahir (2019). The log-odd normal generalized family of distributions with application. Annals of the Brazilian Academy of Sciences 91(2), 1–22.

About the Author G. G. Hamedani, PhD Professor and Editor, JSTA Department of Mathematics, Statistics and Computer Science Marquette University, Milwaukee, Wisconsin, US Email: [email protected]

Professor Hamedani received his PhD from Michigan State University. He is a Professor at Marquette University since 1982. Dr. Hamedani has been Editor of the Journal of Statistical Theory and Applications since 2005 and before that on its first Editorial Board. He is on the Editorial Board of the Journal of Applied Statistical Sciences since its beginning. He is also on the Editorial Board of six other journals. Dr. Hamedani has published over 250 research articles and authored/co-authored four monographs.

Index α-Power Transmuted Generalized Exponen89 tial, 6, 100 Another Two-Parameter Sujatha, 15 Arcsine Exponentiated-X, 11, 202 A Class of Lindley and Weibull, 2, 26 Arcsine Weibull, 8, 146 A Distribution For Instantaneous Failures, 5, 97 Balakrishnan Alpha Skew Normal2 , 9, 169 A Generalization of Sujatha, 15 Benktander Type II, 2, 27 A New, 3, 58 Beta Burr III, 2, 33 Akash, 15 Beta Burr Type X, 14 Alpha Beta Skew Logistic-G, 9, 170 Beta Exponential Pareto, 12, 225 Alpha Logarithm Transmuted Fréchet, 2, 36 Beta Exponentiated Nadarajah-Haghighi, Alpha Logarithmic Transformed Weibull, 14 13, 230 Alpha Power Exponentiated Exponential, 14 Beta Exponentiated Weibull Geometric, 13, Alpha Power Inverse Weibull, 9, 156 230 Alpha Power Inverted Exponential, 9, 156 Beta Generalized Exponentiated Fréchet, 9, Alpha Power Transformation Lomax, 12, 160 212 Beta Generalized Marshall-Olkin Alpha Power Transformation Poisson LindKumarswamy-G, 9, 164 Beta Kumaraswamy Marshall-Olkin-G, 9, ley, 2, 36 163, 166 Alpha Power Transformed Extended ExpoBeta Kumarswamy Burr Type X, 9, 165 nential, 10, 177 Alpha Power Transformed Inverse Lindley, Beta Lindley Geometric, 5, 98 Beta Linear Failure Rate Power Series, 2, 28 11, 196 Alpha Power Transformed Inverse Lomax, Beta Marshall-Olkin Kumarswamy-G, 9, 164 14 Alpha Power Transformed Log-Logistic, 12, Beta Modified Weibull Power Series, 12, 216 224 Alpha Power Transmuted Extended Expo- Beta Odd Lindley-G, 5, 96 Beta Poisson-G, 12, 223 nential, 6, 104 Alpha Skew Generalized Gundermannian, 7, Beta Skew-t, 2, 25 119 Beta Transmuted Weighted Exponential, 4, 73 Alpha-Beta Skew Logistic G, 13 Alpha-Power Generalized Inverse Lindley, Beta Type I Generalized Half Logistic, 11, 11, 205 196 Alpha-Power Pareto, 5, 95 Beta-Complementary Exponential Power Series, 12, 221 Alpha-Power Transformed Lindley, 8, 142 Another Odd Log-Logistic Logarithmic, 5, Beta-G Poisson, 6, 115

466

Index

Bimodal Alpha Skew Logistic G2 , 9, 170 Box-Cox Gamma-G, 15 Burr X Exponential-G, 13 Burr X Exponentiated Lomax, 6, 99 Burr X Fréchet, 7, 136 Burr X Generalized Burr XII, 16 Burr X Nadarajah Haghighi, 7, 136 Burr X Weibull, 12, 225 Burr X-G, 15 Burr X-Kumaraswamy, 6, 113 Burr XII Exponentiated Exponential, 14 Burr XII Exponentiated Weibull, 5, 84 Burr XII Fréchet, 7, 135 Burr XII Inverse Rayleigh, 6, 110 Burr XII Uniform, 12, 209 Burr XII Weibull Logarithmic, 4, 78 Burr-Hatke Exponential, 4, 66 Burr-Hatke Exponentiated Weibull, 4, 61 Burr-Hatke Extended Burr XII, 11, 197 Burr-Hatke Logarithmic BurrXII, 10, 186 Burr-Hatke-G, 4, 61 Burr-Weibull Power Series, 2, 37 Centered

Skew-Normal BirnbaumSaunders, 5, 85 Chen’s Two-Parameter Exponential Power Life-Testing, 2, 20 Chen-G, 10, 181 Combined Exponential-Normal {Generalized Weibull}, 7, 119 Complementary Exponential Geometric, 12, 215 Complementary Exponentiated LomaxPoisson, 12, 216 Composite Generalizers of Weibull, 4, 68 Compound Gamma and Lindley, 4, 74 Cosine-Sine Transformation, 6, 116 Cubic Rank Transmuted Fréchet, 10, 190 Cubic Rank Transmuted Kumaraswamy, 2, 38 Cubic Transmuted, 10, 189 Cubic Transmuted Exponentiated Pareto-1, 14 Cubic Transmuted Gompertz, 13 Cubic Transmuted Pareto, 3, 8, 57, 141

Cubic Transmuted Power Function, 10, 182 Cubic Transmuted Uniform, 6, 115 Cubic Transmuted Weibull, 2, 38 Double Truncated Transmuted Fréchet, 10, 185 Doubly Truncated Extreme Value Type I, 9, 171 Dual Exponentiated Weibull, 7, 137 erf-G, 13 Exponential Skew-Normal, 15 Exponential Transmuted Fréchet, 8, 148 Exponential-Gamma, 11, 202 Exponentiated Additive Weibull, 13, 230 Exponentiated Burr XII Power Series, 4, 77 Exponentiated Cubic Transmuted Exponential, 11, 206 Exponentiated Exponential Lomax, 10, 191 Exponentiated Garima, 14 Exponentiated Generalized Exponentiated Exponential, 2, 24 Exponentiated Generalized Extended Gompertz, 4, 64 Exponentiated Generalized Extended Pareto, 1, 2, 16, 26 Exponentiated Generalized Inverse Rayleigh, 3, 44 Exponentiated Generalized Inverted Exponential, 2, 23 Exponentiated Generalized Pareto, 3, 46 Exponentiated Generalized Power Function, 9, 162 Exponentiated Generalized Power Lindley, 8, 155 Exponentiated Generalized Power Series, 6, 117 Exponentiated Generalized Standardized Gumbel, 11, 199 Exponentiated Gumbel Exponential, 9, 157 Exponentiated Half-Logistic Lomax, 15 Exponentiated Inverse Rayleigh, 4, 72 Exponentiated Kumaraswamy-Weibull, 4, 65 Exponentiated Kumarsawamy-G, 3, 58

Index Exponentiated Length Biased Exponential, 11, 205 Exponentiated Log-Sinh Cauchy, 6, 115 Exponentiated Mukherjee-Islam, 3, 55 Exponentiated Nadarajah Haghighi Poisson, 7, 123 Exponentiated Negative Binomial, 66 Exponentiated New Weighted Weibull, 4, 72 Exponentiated Odd Chen-G, 13 Exponentiated Odd Log-Logistic Normal, 5, 95 Exponentiated Odd Log-Logistic-G, 5, 82 Exponentiated Poisson-Exponential, 14 Exponentiated Power Function, 14 Exponentiated Power Generalized Weibull, 2, 29 Exponentiated Power Generalized Weibull Power Series, 12, 214 Exponentiated Shanker, 14 Exponentiated Topp-Leone, 5, 80 Exponentiated Transmuted Length-Biased Exponential, 14 Exponentiated Transmuted Power Function, 3, 43 Exponentiated Transmuted Weibull Geometric, 12, 215 Exponentiated Truncated Inverse WeibullGenerated, 11, 208 Exponentiated Two Parameter Pranav, 14 Exponentiated Weibull Weibull, 6, 109 Exponentiated Weibull-Exponentiated Weibull, 8, 152 Exponentiated Weighted Exponential, 2, 19 Exponentiated-Epsilon, 10, 183 Exponentiated-Exponential Logistic, 4, 67 Extended Beta Power Function, 11, 202 Extended Enlarg Transmuted Exponential, 3, 41 Extended Exp-G, 15 Extended Exponentiated Weibull, 12, 219 Extended Generalized Inverse Exponential, 13 Extended Generalized Lindley, 14 Extended Log-Logistic, 8, 148 Extended New Generalized Exponential, 7,

467

124 Extended Normal, 6, 99 Extended Odd Fréchet-G, 2, 5, 35, 95 Extended Odd Log-Logistic, 10, 184 Extended Odd Weibull Exponential, 9, 172 Extended Odd Weibull-G, 15 Extended Poisson Fréchet, 8, 153 Extended Poisson Lomax, 16 Extended Power Lindley-G, 8, 150 Extended Pranav, 7, 127 Extended Weibull-G, 4, 68 Extended Weighted Exponential, 1, 16 Flexible Additive Weibull, 10, 188 Flexible Weibull Burr XII, 16 Fréchet Weibull, 12, 224 Functional Weighted Exponential, 3, 50 G-Fixed-Topp-Leone, 12, 209 Gamma Burr Type X, 14 Gamma Dual Weibull, 7, 122 Gamma Generalized Normal, 2, 32 Gamma Inverse Weibull, 14 Gamma Kumarswamy-G, 15 Gamma Power Half-Logistic, 15 General Class, 3, 40 General Transmuted Family, 3, 54 Generalization of Exponential and Lindley, 13 Generalization of Two-Parameter Lindley, 3, 51 Generalized Burr X-G, 1, 17 Generalized Burr XII Power Series, 6, 116 Generalized Class, 12, 221 Generalized DUS Lindley, 11, 201 Generalized Extended Exponential-Weibull, 6, 114 Generalized Extended Inverse Weibull, 6, 108 Generalized Gamma Burr III, 2, 25 Generalized Gamma Exponentiated Weibull, 11, 207 Generalized Gamma-G, 14 Generalized Gamma-Generalized Inverse Weibull, 15

468

Index

Generalized Gompertz-Generalized Gompertz, 6, 108 Generalized Gudermannian, 5, 7, 82, 119 Generalized Inverse Lindley, 8, 149 Generalized Inverse Lindley Power Series, 8 Generalized Inverse Marshall-Olkin, 10, 176 Generalized Inverse Pareto-G, 7, 133 Generalized Inverse Weibull- Generalized Inverse Weibull, 5, 97 Generalized Inverse Weibull-Generalized Inverse Weibull, 4, 77 Generalized Inverted Kumaraswamy, 2, 26 Generalized Inverted Kumarswamy Generated, 15 Generalized Kumaraswamy-G, 8, 155 Generalized Lindley, 10, 175 Generalized Lindley Power Series, 1, 18 Generalized Log-Lindley, 4, 62 Generalized Marshall-Olkin Extended Burr XII, 8, 146 Generalized Marshall-Olkin Extended BurrIII, 8, 140 Generalized Marshall-Olkin Inverse Lindley, 14 Generalized Marshall-Olkin Poisson-G, 13 Generalized Marshall-Olkin Transmuted-G, 13 Generalized Modified Exponential-G, 9, 170 Generalized Moment Exponential Power Series, 9, 169 Generalized New Extended Weibull, 12, 210 Generalized Odd Burr III-G, 3, 48 Generalized Odd Fréchet-G, 3, 49 Generalized Odd Generalized Exponential Fréchet, 16 Generalized Odd Generalized Exponential G, 10, 184 Generalized Odd Half-Cauchy-G, 8, 155 Generalized Odd Inverted Exponential-G, 12, 214 Generalized Odd Lindley-G, 7, 128 Generalized Odd Linear Exponential, 13 Generalized Odd Log-Logistic Exponential, 3, 9, 42, 172 Generalized Odd Log-Logistic Log-Normal,

14 Generalized Odd Log-Logistic-G, 10, 187 Generalized Odd Log-Logistics Inverse Weibull, 5, 85 Generalized Odd Lomax Generated, 6, 111 Generalized Raised Cosine, 15 Generalized Reciprocal Exponential, 14 Generalized Transmuted Fréchet, 2, 5, 27, 86 Generalized Transmuted GompertzMakeham, 3, 44 Generalized Transmuted Moment Exponential, 14 Generalized Transmuted Poisson-G, 13 Generalized Transmuted Power Function, 3, 55 Generalized Uniform, 10, 174 Generalized Weibull Uniform, 11, 209 Generalized Weighted Exponential, 14 Geometric Lindley Poisson 1, 6, 106 Gompertz Exponential, 15 Gompertz Extended Generalized Exponential, 10, 184 Gompertz Flexible Weibull, 12, 226 Gompertz Fréchet, 10, 180 Gompertz Length Biased Exponential, 9, 162 Gompertz Lomax, 2, 20 Gompertz-Alpha Power Inverted Exponential, 10, 183 Gumbel-Burr XII, 9, 163 Half-Logistic XGamma, 13 Hamza, 15 Harris Extended Lindley, 4, 69 Hyperbolic Cosine Weibull, 13 Hyperbolic Sine Rayleigh, 4, 71 Intervened Binomial Compound, 10, 179 Intervened Geometric Compound, 10, 178 Intervened Negative Binomial Compound, 10, 178 Intervened Poisson Compound, 10, 179 Inverse Gompertz, 4, 71 Inverse Lomax-G, 15 Inverse Power Lomax, 4, 64

Index Inverse Weibull Generator, 1, 17 Inverse Weibull Geometric, 6, 112 Inverse Weibull Poisson, 6, 112 Inverse Weighted Lindley, 3, 47 Inverse XGamma, 8, 141 Inverted Beta, 3, 56 Inverted Beta Lindley, 3 Inverted Modified Lindley, 15 Inverted Nadarajah-Haghighi, 2, 31 Inverted Weighted Exponential, 2, 20 Jamal Logistics-X, 3, 45 Jamal Weibull-X, 3, 11, 45, 195 Janardan-Power Series, 4, 63 Kumaraswamy Alpha Power Inverted Exponential, 9, 159 Kumaraswamy Extension Exponential, 3, 41 Kumaraswamy Flexible Weibull, 2, 22 Kumaraswamy Generalized Linear Exponential, 4, 59 Kumaraswamy Half-Logistics, 2, 21 Kumaraswamy Marshall-Oklin Exponential, 2, 21 Kumaraswamy Moment Exponential, 9, 168 Kumaraswamy Odd Lindley-G, 7, 138 Kumaraswamy Weibull, 2, 21 Kumaraswamy-Chen, 2, 22 Kumaraswamy-XTG, 2, 22 Kumarsawamy Marshall-Olkin Modified Weibull, 7, 133 Kumarswamy Alpha Power-G, 16 Kumarswamy Exponentiated U-Quadratic, 7, 126 Kumarswamy Log-Logistic Weibull, 9, 156 Kumarswamy Poisson-G, 12, 211 Kumarswamy Reciprocal, 10, 177 Kumarswamy Sushila, 14 Kumarswamy Type I Half Logistic, 6, 110 L-Logistic, 7, 132 Leaned Normal, 7, 128 Length Biased XGamma, 7, 131 Length-Biased Suja, 11, 205 Lindley Negative-Binomial, 15 Lindley Quasi XGamma, 11, 203

469

Lindley Rayleigh, 11, 208 Lindley Weibull, 2, 29 Linearly Decreasing Stress Weibull, 7, 132 Log-Balakrishnan-Alpha-Skew-Normal, 10, 188 Log-Beta Modified Weibull, 12, 227 Log-Epsilon-Skew Normal, 14 Log-Gamma-Generated, 13 Log-Odd Log-Logisticc BirnbaumSaunders-Poisson, 5, 96 Log-Odd Logistic-Weibull, 8, 142 Log-Odd Normal Generalized, 8, 149 Log-Sinh Cauchy Promotion, 4, 58 Log-Weighted Exponential, 15 Log-Weighted Pareto, 10, 177 Logarithm Transformed Lomax, 9, 162 Logarithmic Kumarswamy, 5, 86 Logarithmic Transformed Inverse Weibull, 16 Logistic Exponential, 14 Lomax D function Generalized Weibull, 13 Lomax Exponential, 9, 171 Lomax Exponentiated Weibull, 10, 178 Lomax Gompertz-Makeham, 8, 147 Lomax Inverse Weibull, 15 Lomax Weibull, 8, 144 Lomax-Lindley, 6, 108 Marshall-Olkin Alpha Power, 2, 30 Marshall-Olkin Alpha Power Inverse Exponential, 6, 117 Marshall-Olkin Burr Exponential-2, 12, 214 Marshall-Olkin Burr X, 4, 76 Marshall-Olkin Exponential Gompertz, 11, 207 Marshall-Olkin Extended Exponential, 6, 103 Marshall-Olkin Extended Flexible Weibull, 4, 78 Marshall-Olkin Extended Power Function, 3, 42 Marshall-Olkin Extended Power Lomax, 13 Marshall-Olkin Extended Quasi Lindley, 8, 145

470

Index

Marshall-Olkin Extended Weibull Exponential, 8, 148 Marshall-Olkin Generalized Burr XII, 8, 149 Marshall-Olkin Generalized Pareto, 10, 188 Marshall-Olkin Generated Gamma, 2, 31 Marshall-Olkin Inverse-Lomax, 13 Marshall-Olkin Lehmann Burr X, 15 Marshall-Olkin Length Biased Exponential, 9, 168 Marshall-Olkin Modified Burr III, 7, 130 Marshall-Olkin Odd Burr III-G, 15 Marshall-Olkin Odd Lindley G, 5, 86 Marshall-Olkin Power Generalized Weibull, 13 Marshall-Olkin Topp Leone-G, 14 Marshall-Olkin Transmuted-G, 15 McDonald Gumbel, 13 McDonald Modified Burr-III, 10, 174 Minimum Guarantee Lindley, 3, 56 Minimum Gumbel Burr, 15 Minimum Weibull-Burr, 15 Mixture Pareto Log-Gamma, 14 Modi Generator, 14 Modified Beta Generalized Linear Failure Rate, 12, 15, 222 Modified Beta Gompertz, 8, 151 Modified Beta Linear Exponential, 8, 140 Modified Beta Modified-Weibull, 4, 59 Modified Burr XII, 9, 173 Modified Extended Generalized Exponential, 7, 123 Modified Fréchet, 6, 115 Modified Generalized Marshall-Olkin, 7, 120 Modified Kies Generalized, 15 Modified Odd Weibull-G, 10, 174 Modified Slashed Half-Normal, 14 Modified T-X, 10, 185 Modified Weibull-G, 4, 74 Mrashall-Olkin Kappa, 9, 157 Muth Generated, 2, 33

Nasir Logistic-X, 3, 45 Nasir Weibull-Generalized, 3, 46 New Alpha-Power Transformation, 3, 50 New Beta Power Transformed, 11, 12, 207, 220 New Cubic Rank Transmutation, 14 New Exponential Trigonometric, 12, 210 New Extended Alpha Power Transformed, 5, 82 New Extended Burr III, 11, 195 New Extended Generalized Burr III, 4, 76 New Extended Weibull, 14 New Extended-F, 14 New Family of Heavy Tailed, 12, 219 New Generalization of Weibull-Exponential, 6, 103 New Generalized Akash, 13, 228 New Generalized Rayleigh, 13 New Generalized Transmuted Inverse Exponential, 7, 126 New Heavy Tailed Family of Claim, 12, 219 New Libby-Novick, 9, 173 New Lifetime Exponential-Weibull, 14 New Lindley Exponential, 5, 88 New Mixture of Exponential-Gamma, 12, 228 New Modified Burr III, 15 New Odd Generalized ExponentialExponential, 3, 40 New Odd Log-Logistic Chen, 13, 229 New Odd Log-Logistic Half-Logistic, 5, 93 New Power Topp-Leone Generated, 8, 151 New Three Parameter Paralogistic, 4, 63 New Two-Parameter Sujatha, 15 New Unit-Lindley, 13 New Weighted Exponential, 2, 19 New Weighted Transmuted Exponential, 12, 211 Normal Generalized Hyperbolic Secant, 7, 133 Normal-C, 8, 150 Normal-Poisson, 12, 218

Nadarajah Haghighi Topp Leaone-G, 8, 152 Nadarajah-Haghigh Geometric, 6, 114 Odd Birnbaum-Saunders, 7, 122 Nadarajah-Haghighi Lindley, 6, 114 Odd Burr Generalized Rayleigh, 15

Index Odd Burr III Exponential, 14 Odd Burr III G-Negative Binomial, 3, 50 Odd Burr III Weibull, 4, 70 Odd Burr- Generalized, 5, 90 Odd Burr-G Poisson, 3, 54 Odd Dagum, 11, 200 Odd Exponential-Pareto IV, 12, 213 Odd Exponentiated Half-Logistic Exponential, 14 Odd Fréchet Inverse Exponential, 6, 107 Odd Fréchet Inverse Lomax, 14 Odd Fréchet Inverse Rayleigh, 6, 105 Odd Fréchet Inverse Weibull, 6, 104 Odd Gamma Weibull-Geometric, 15 Odd Generalized Exponential Power Function, 6, 111 Odd Generalized Exponential Type-I Generalized Half Logistic, 7, 136 Odd Generalized Exponential-Exponential, 9, 159 Odd Generalized Exponentiated Inverse Lomax, 13 Odd Generalized Gamma-G, 15 Odd Hyperbolic Cosine KG, 5, 81 Odd Inverse Pareto-Exponential, 10, 186 Odd Inverse Pareto-G, 6, 105 Odd Lindley Exponentiated Weibull, 2, 4, 11, 35, 73, 193 Odd Lindley Fréchet, 3, 52 Odd Lindley Half Logistics, 12, 222 Odd Lindley Inverse Exponential, 13 Odd Lindley Lomax, 6, 99 Odd Lindley-Rayleigh, 13 Odd Log-Logistic Dagum, 11, 199 Odd Log-Logistic Exponential Gaussian, 10, 180 Odd Log-Logistic Exponentiated Gumbel, 4, 69 Odd Log-Logistic Exponentiated Weibull, 5, 89 Odd Log-Logistic Generalized Gompertz, 5, 93 Odd Log-Logistic Generalized HalfNormal, 5, 98 Odd Log-Logistic Generalized Half-Normal

471

Poisson, 4, 60 Odd Log-Logistic Generalized Inverse Gaussian, 2, 39 Odd Log-Logistic Geometric Normal, 4, 78 Odd Log-Logistic Geometric-G, 8, 145 Odd Log-Logistic Lindley-G, 8, 154 Odd Log-Logistic Log-Normal, 5, 92 Odd Log-Logistic Marshall-Olkin Generalized Half-Normal, 5, 94 Odd Log-Logistic Marshall-Olkin Power Lindley, 5, 79 Odd Log-Logistic Poisson-G, 5, 88 Odd Log-Logistic Topp-Leone G, 5, 91 Odd Lomax Exponential, 11, 194 Odd Lomax Fréchet, 12, 225 Odd Lomax-G, 2, 28 Odd Weibull, 4, 75 Odoma, 134 One Parameter Polynomial Exponential-G, 14 One-Parameter Weibull-Type, 5, 96 Perturbed Half-Normal, 12, 218 Poisson Burr Type X Log-Logistic, 3, 40 Poisson Burr X -Fréchet, 8, 154 Poisson Burr X Burr XII, 9, 166 Poisson Burr X Generalized Lomax, 9, 167 Poisson Burr X Inverse Rayleigh, 10, 176 Poisson Burr X Pareto Type II, 7, 135 Poisson Burr X Weibull, 6, 102 Poisson Exponential-G, 12, 217 Poisson Exponentiated Erlang-Truncated Exponential, 3, 56 Poisson Nadarajah-Haghighi, 6, 102 Poisson Odd Generalized Exponential, 10, 173 Poisson Rayleigh Burr XII, 11, 201 Poisson Rayleigh Generalized Lomax, 10, 187 Poisson Rayleigh Log-Logistic, 8, 145 Poisson Topp-Leone Inverse Weibull, 3, 54 Poisson-X, 6, 100 Poly-Exponential Transformation, 12, 211 Power Binomial Exponential 2, 2, 33 Power Function Power Series, 4, 65

472

Index

Power Gompertz, 8, 143 T-R {Y} Power Series, 13, 228 Power Lindley Generated, 4, 70 Three Parameter Generalized Lindley, 2, 34 Power Lindley Geometric, 11, 194 Topp-Leone Compound Rayleigh, 7, 137 Power Log-Dagum, 12, 210 Topp-Leone Exponentiated-G, 11, 195 Power Muth, 10, 182 Topp-Leone Generalized Inverted KumarPower-Exponential Hazard Rate, 4, 64 swamy, 4, 75 Power-Linear Hazard Rate, 5, 94 Topp-Leone Generated q-Exponential, 5, 81 Product of Two Independent Weibull and Topp-Leone Generator, 5, 81 Lindley, 15 Topp-Leone Inverse Weibull, 5, 79 Topp-Leone Lindley, 16 Quasi Sujatha, 11, 204 Topp-Leone Lomax, 5, 80 Quasi XGamma-Geometric, 7, 129 Topp-Leone Marshall-Olkin-G, 12, 223 Quasi XGamma-Poisson, 7, 122 Topp-Leone Mukherjee-Islam, 3, 52 Topp-Leone Odd Lindley-G, 5, 91 Raised Cosine, 8, 139 Topp-Leone Odd Log-Logistic Exponential, Ratio Exponentiated General, 15 7, 121 Ratio of Two Independent Weibull and LindTopp-Leone Rayleigh, 14 ley, 15 Topp-Leone Weibull-Lomax, 15 Rayleigh Rayleigh, 10, 181 Topp-Leone Weighted Weibull, 4, 61 Rayleigh-Geometric, 14 Topp-Leone-G Poisson, 6, 104 Reduced New Modified Weibull, 4, 67 Topp-Leone-Lomax, 9, 159 Reflected Power Function, 13 Transmuted Alpha Power Inverse Rayleigh, Right Truncated Power Lomax, 13 11, 206 Right Truncated-X, 14 Transmuted Alpha Power-G, 14 Risti´c-Balakrishnan Extended Exponential, Transmuted Arcsine, 9, 166 11, 198 Transmuted Burr Type X, 11, 193 Ristic-Balakrishnan Odd Log-Logistic-G, Transmuted Complementary Exponential 14 Power, 13 Transmuted Exponential-G, 12, 227 Sine Kumarswamy-G, 15 Transmuted Exponentiated Additive Sinh Inverted Exponential, 15 Weibull, 1, 18 Size Biased Gamma Lindley, 9, 161 Transmuted Exponentiated U-quadratic, 2, Skew Scale Mixtures Normal, 15 24 Skew t-Distribution of Three Degrees of Transmuted Exponentiated Weibull, 8, 139 Freedom, 9, 161 Transmuted Extended Exponential, 6, 117 Skewed Generalized Logistic, 8, 142 Transmuted Extended Lomax, 7, 121 Slash Lindley-Weibull, 11, 198 Transmuted Four Parameters Generalized Slash Maxwell, 10, 185 Log-Logistic, 6, 112 Slash Power Maxwell, 14 Transmuted General, 15 Slashed Power-Lindley, 3, 46 Transmuted Generalized Extreme Value, 11, Slashed Quasi-Gamma, 14 192 T-Burr, 15 Transmuted Generalized Gamma, 6, 113 T-Dagum{Y }, 213 Transmuted Generalized Inverted ExponenT-Dagum{Y }, 12 tial, 9, 160 T-Kumarswamy, 12, 217 Transmuted Generalized Lindley, 13, 229

Index Transmuted Generalized Linear Exponential, 2, 22 Transmuted Generalized Odd Generalized Exponential-G, 4, 62 Transmuted Generalized Power Weibull, 9, 161 Transmuted Half Logistic, 7, 125 Transmuted Half Normal, 10, 11, 190, 191 Transmuted Ishita, 11, 192 Transmuted Kumaraswamy Lindley, 13, 231 Transmuted Lomax Exponential, 7, 127 Transmuted Lomax-G, 11, 194 Transmuted Modified Weibull, 10, 189 Transmuted New Weibull Pareto, 8, 144 Transmuted Odd Fréchet-G, 13 Transmuted Power Function, 9, 167 Transmuted Power Gompertz, 15 Transmuted Singh-Maddala, 7, 125 Transmuted Topp Leone Exponentiated Fréchet, 7, 118 Transmuted Topp-Leone Weibull, 16 Transmuted Transmuted-G, 2, 32 Transmuted Type I Generalized Logistic, 8, 138 Transmuted Type II Generalized Logistic, 12, 220 Transmuted Weibull, 10, 189 Truncated Burr-G, 12, 212 Truncated Cauchy Power-G, 11, 198 Truncated Discrete Linnik Weibull, 8, 139 Truncated Exponential Skew Logistic, 9, 169 Truncated Inverted Kumaraswamy Generated, 8, 147 Truncated Weibull Fréchet, 12, 223 Truncated Weibull Power Lomax, 6, 107 Truncated-Logistic Skew-Symmetric, 7, 119 Two-Parameter Sujatha, 15 Type I Half Logistic Power Lindley, 3, 49 Type I Half-Logistic, 5, 90 Type I Half-Logistic Burr X, 6, 106 Type I Half-Logistic Exponential, 3, 51 Type I Half-Logistic Modified Weibull, 11, 200 Type I Half-Logistic Rayleigh, 12, 227

473

Type I New Heavy Tailed Weibull, 8, 146 Type II Exponentiated Half Logistic Generated, 7, 130 Type II General Inverse Exponential, 15 Type II Generalized Topp-Leone-G, 5, 83 Type II Half Logistic Exponentiated Exponential, 11, 203 Type II Half Logistic Ibrahim, 7, 120 Type II Half Logistic Weibull, 3, 43 Type II Kumaraswamy Half LogisticGenerated, 2, 39 Type II Power Topp-Leone Generated, 9, 163 Type II Topp-Leone Generated, 4, 72 Type II Topp-Leone Inverted Kumaraswamy, 8, 151 Type II Topp-Leone Power Ishita, 13 Type II Topp-Leone Power Lomax, 9, 13, 158 Unit Generalized Half Normal, 13 Unit Johnson SU , 16 Unit Modified Burr-III, 13 Unit Nadarajah-Haghighi Generated, 13 Unit- Birnbaum-Saunders, 13 unit-Improved Second-Degree Lindley, 7, 138 Unit-Inverse Gaussian, 3, 47 Unit-Lindley, 11, 199 Unit-Marshall-Olkin Extended Exponential, 3, 51 Unit-Weibull, 11, 201 Weibull Alpha Power Inverted Exponential, 10, 182 Weibull Burr Type X, 15 Weibull Burr XII, 15 Weibull Exponentiated Exponential, 6, 14, 101 Weibull Generalized Burr XII, 8, 153 Weibull Generalized Exponentiated Weibull, 5, 87 Weibull Generalized Log-Logistic, 4, 62 Weibull Marshall-Olkin Lindley, 10, 175 Weibull Pareto, 3, 46 Weibull-G Power Series, 2, 34

474

Index

Weibull-Inverse Lomax, 6, 9, 109, 158 Weibull-Lindley, 2, 34 Weibull-Lomax, 3, 52 Weibull-Moment Exponential, 3, 47 Weibull-Negative Binomial, 13 Weighted Exponential, 15 Weighted Exponential Gompertz, 10, 181 Weighted Garima, 10, 190 Weighted Inverse Gamma, 4, 59 Weighted Inverse Lévy, 11, 204 Weighted Inverse Nakagami, 13 Weighted Inverted Weibull, 12, 213 Weighted Ishita, 11, 203 Weighted Modified Weibull, 6, 101 Weighted Nakagami, 11, 204 Weighted Power Lindley, 13 Weighted T-X, 8, 143 Weighted Version of Generalized Inverse Weibull, 11, 204 Weighted XGamma, 7, 131 Wibull Inverse Lomax, 7, 134 Wrapped Lindley, 11, 197 Wrapped Lindley-Exponential, 7, 134 Wrapped XGamma, 7, 132 X Gamma Weibull, 4, 67 XGamma, 7, 129 XGamma-G, 9, 158 Zero Spiked Gamma Weibull, 2, 30 Zero Truncated Poisson Topp Leone Fréchet, 11, 196 Zero Truncated Poisson Topp Leone Weibull, 3, 5, 57, 83 Zero Truncated Poisson Topp-Leone Burr XII, 3, 53 Zero Truncated Poisson Topp-Leone Exponentiated Weibull, 3, 53 Zero-Truncated Poisson Exponentiated Gamma, 8, 143 Zero-Truncated Poisson-Power Function, 8, 141 Zografos Balakrishnan Power Lindley, 12, 217 Zografos-Balakrishnan Burr XII, 9, 165 Zografos-Balakrishnan Fréchet, 2, 37

Zografos-Balakrishnan Lindley-Poisson, 5, 96 Zografos-Balakrishnan Odd Log-Logistic Generalized Half-Normal, 5, 94 Zubair-G, 221 Zubair-G, 12 Zubair-Inverse Lomax, 15