New Trends in Probability and Statistics. Vol. 5 Multivariate Statistics: Proceedings of the 6th Tartu Conference, Tartu, Estonia, 19–22 August 1999 [Reprint 2011 ed.] 9783110944655, 9783110628173

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New Trends in Probability and Statistics - Volume 5 Multivariate Statistics

NEW TRENDS IN PROBABILITY AND STATISTICS Volume 5 Multivariate Statistics Proceedings of the 6th Tartu Conference, Tartu, Estonia, 19-22 August 1999

Editors T. Kollo, Ε.-Μ. Tiit and Μ. Srivastava

///VSP/// VILNIUS, LITHUANIA

UTRECHT, THE NETHERLANDS TOKYO, JAPAN

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

TEV Ltd. Akademijos 4 2600 Vilnius Lithuania

©VSP BV & TEV Ltd. 2000

First published 2000

ISBN 90-6764-329-7 (VSP) ISBN 9986-546-85-0 (TEV)

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Typeset in Lithuania by TEV Ltd., Vilnius Printed in Lithuania by Vilspa, Vilnius

CONTENTS Preface vii Estimation in the extended linear model F. Akdeniz

1

Discrete optimization of statistical sample sizes in simulation by using the hierarchical resampling method A. Andronov and M. Fioshin

9

Kernel-type estimation of bivariate distribution function for associated random variables C. Azevedo and P. E. Oliveira

17

On local dependence function for multivariate distributions I. Bairamov and S. Kotz

27

The tuning constant problem in regression type models J. A. Branco and M. M. Souto de Miranda

45

The usefulness of uniformity in experimental design K.-T. Fang and Ch. -X. Ma

51

Approximation of distributions by sphere M. Käärik

61

Econometric modeling of the Lithuanian economy Ζ Kalinauskas

67

Non-null distribution of Läuter's F-statistic: A simulation study T. Kollo and Α.-Μ. Parring

77

Zero-boundary Voronoi partitions J. Lember

87

On expected values of fourth-degree matrix products of a multinormal matrix variate H. Neudecker

97

A multivariate Buckley-James estimator U. Pötter

117

A new algorithm of the linear discriminant function using integer programming S. Shinmura

133

Robustification of "approximating approach" in simultaneous equations models S. Staleuskaya and Yu. Kharin

143

Data compression and statistical inference H. Strasser

151

Estimation the poverty line as a problem of multivariate statistics E.-B. TiitandE. Käärik

173

Dependence structure of a multivariate distribution characterised by the correlation matrix E.-M. Tiit, V. Puusepp and H.-L. Viirsalu

179

Approximation of multivariate probability functions by empirical marginal distributions and dependence measures E. -M. Tiit and M. Vähi

187

Sampling design as a multivariate distribution I. Traat

195

Class of sampling designs with given marginals I. Traat and A. Kukk

209

A remark on the extended growth curve model I. Zezula

219

PREFACE The tradition of holding Conferences in Multivariate Statistics at the University of Tartu began in 1977 when the first one was held. The participants in the first four conferences (held in 1977,1981,1985 and 1989) were mostly from the Soviet Union and the language of the Conference was also Russian. The Fifth Tartu Conference held in 1994 opened the doors for other participants of the world and it became truly international. It was attended by 70 participants from 18 countries. In 1997 the Organising Committee of the 52nd Meeting of the International Statistical Institute in Helsinki proposed to hold a Satellite Conference in Multivariate Statistics in Tartu. The idea was enthusiastically greeted by the staff of the Institute of Mathematical Statistics at the University of Tartu and by the Estonian Statistical Society. The Sixth International Conference in Multivariate Statistics was organised under the auspices of the Bernoulli Society and the Association of Statistical Computing. Professor Timo Teräsvirta from Stockholm was the Chair of the Programme Committee (Bernoulli Society) with its active members Hans-Herman Bock, Dietrich von Rosen and Albert Satorra. The Sixth Tartu Conference in Multivariate Statistics was held at the University of Tartu between August 19 and August 22,1999. There were more than 60 participants from 19 countries. The Programme Committee helped select the speakers for the Conference. Active participation of the honourable keynote lecturer T. W. Anderson and distinguished invited speakers T. Durbin, Κ. T. Fang, S. Johansen, J. Läuter, Η. Neudecker, Μ. S. Srivastava and H. Strasser created stimulative working atmosphere at the conference. It was decided to publish the presented results in the Proceedings of the Conference. The volume contains 21 papers from 30 authors. The Editors are extremely thankful to the authors for their contributions and to the numerous referees for their seemingly invisible, but important work. The papers were remarkably improved due to their valuable suggestions and remarks. The papers are presented in the Proceedings in alphabetic order. During the editing obvious misprints have been corrected, but the style of the authors is unchanged and responsibility for the presentation of the material remains to the authors. We hope that the present set of papers passes onto the readers the stimulating and creative atmosphere of the 6th Tartu Conference on Multivariate Statistics and encourages them to participate in the next Tartu Conference in the new millennium. T. Kollo, E.-M. Tiit and M. Srivastava, editors

New Trends in Prob, and Stat., Vol. 5, pp. 1-7

T. Kollo etal. (Eds) © 2000 VSP/TEV

ESTIMATION IN THE EXTENDED LINEAR MODEL FIKRIAKDENIZ Department of Mathematics, Faculty of Arts and Sciences, £ukurova University, 01330 Adana, Turkey e-mail: [email protected]

ABSTRACT The purpose of this paper is to apply the theory of the extended linear model to a model developed by Mandel (1961). The Mandel's model has been analysed and some properties of the estimators of X ß are given.

1. INTRODUCTION Let y = X/J + e

(1)

be any linear model where y is a η χ 1 vector of observations; X is a η χ ρ known fixed matrix of rank r ^ min(n, p)\ β is a ρ χ 1 unknown fixed vector, e is an η χ 1 unobserved random normal vector with E(e) = 0 and E(ee') = σ£ΐ. The linear model (1) may, at times, be inadequate in its description of the experimental situation. The linear model may be extended by including nonlinear terms which are known functions of β or β. The extended linear model (ELM) was defined by Milliken and Graybill (1970). The extended linear model y = Xß + F a + e,

(2)

where F = [ f i j (·)] is an η χ k matrix and the functional form of frj (·) is known but is a function of the estimable functions of β\ thus we can write it as f j (Xß). Also α is an unknown k χ 1 vector of additional parameters. As it now stands, F is a matrix with unknown elements since Xß is unknown. It is of interest to determine if the extended linear model is appropriate in describing the experimental situation. Akdeniz and Milliken (1975) defined the conditional extended linear model and developed the procedure to test a general linear hypothesis involving nonlinear parameters. This paper consists of an application of the theory of the extended linear model to a model developed by Mandel (1961). In addition, the invariance of some specific matrix expression with respect to the choice of generalized inverses is given. Our interest focuses on the connections between the ordinary least squares estimator ( O L S E ) and the best linear unbiased estimator (BLUE) of Xß.

F. Akdeniz

2

2. TWO-WAY CLASSIFICATION MODEL WITH INTERACTION There are several examples of problems fitting into the framework of the ELM that involve a model with a linear part and a nonlinear part involving this linear segment. We shall illustrate models (1) and (2) with a well known example. Consider a twoway classification model with no interaction. The model corresponding to (1) is yij = μ + τ,· + Sj + e;;·,

i = 1 , 2 , . . . , f; j = 1 , 2 , . . . , b.

(3)

Though the investigator may feel this model is unsatisfactory, it may be useful to determine if there is a τδ interaction; i.e., he may want to make a preliminary examination of the "more complete moder yij =μ + η+ 8j + (τδ)α +eu, t Σ τ,· = 0,

£

ι=1

j=1 b Σ(τδ)α

i = 1 , 2 , . . . , r; j = 1,2

b Sj = 0,

= 0,

b,

(4)

t Σ > ό ) , 7 = 0, 1=1

e,j ~ NID(0,