Statistical Inference in Financial and Insurance Mathematics with R (Optimization in Insurance and Finance Set) [1 ed.] 1785480839, 9781785480836

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Statistical Inference in Financial and Insurance Mathematics with R

Optimization in Insurance and Finance Set coordinated by Nikolaos Limnios and Yuliya Mishura

Statistical Inference in Financial and Insurance Mathematics with R

Alexandre Brouste

To Sophie, Aurèle and Lucie

First published 2018 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Press Ltd 27-37 St George’s Road London SW19 4EU UK

Elsevier Ltd The Boulevard, Langford Lane Kidlington, Oxford, OX5 1GB UK

www.iste.co.uk

www.elsevier.com

Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. For information on all our publications visit our website at http://store.elsevier.com/ © ISTE Press Ltd 2018 The rights of Alexandre Brouste to be identified as the authors of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress ISBN 978-1-78548-083-6 Printed and bound in the UK and US

Preface

This book summarizes lectures which I have given for eight years in a Master course in Statistics for Finance and Insurance at Le Mans University and the Institute of Risk and Insurance. This book is thus dedicated to students, engineers and practitioners interested in mathematical statistics and parametric estimation in general. Our statistics team is historically focused on statistics for stochastic processes and their applications. Statistical inference for continuously observed stochastic processes has mainly been developed here by Yury Kutoyants for Markov processes and Marina Kleptsyna for long-memory processes. I hope this book can contribute to this local story. I would also like to thank Alison Bates, Christophe Dutang, Didier Loiseau, Mathieu Rosenbaum and Alexandre Popier for taking some of their precious time to carry out a careful revision. I am equally grateful to all the other people who support me in my everyday life. When I was young, my grandfather Charles Brouste offered me a book that I had chosen randomly on the little mathematical shelf of a small bookshop in Pau, France. This book was a lecture on mathematical statistics by Alain Monfort and it has never left me. A love of mathematics (more especially statistics) and fond memories of the pleasure felt when I managed

x

Statistical Inference in Financial and Insurance Mathematics with R

to solve the problems given to me by my grandfather in summer times also never left me. If one person reading this book chooses to work harder in mathematical statistics, it will not have been written in vain. Alexandre B ROUSTE October 2017

List of Notations – A is the indicator function of A, i.e. x ∈ A or x ∈ / A.

A (x)

= 1 or 0 accordingly as

– ∗ is the transposition. – B is the σ-algebra of Borel sets of

.

– ◦ is the composition of two applications. –C(

+,

) continuous function defined on

– δa is the Dirac measure at a ∈ B∈B .

+

and valued in

on B defined by δa (B) =

. B (a)

for

– ∇ is the gradient (nabla symbol). – h · ν : the probability measure h · ν is defined by ˆ h · ν(A) = h(x)ν(dx). A

– Ip is the p × p identity matrix. – –

is the set of positive integers. ∗

is the set of strictly positive integers.

– N (μ, σ 2 ) is the distribution of a Gaussian random variable of mean μ and variance σ 2 . –

is the real line.

xii

Statistical Inference in Financial and Insurance Mathematics with R

–

+

is the set of positive real numbers.

–

∗ +

is the set of strictly positive real numbers.

–

d

is the d-dimensional Euclidean space.

– σ (Yu , 0 ≤ u ≤ s) is the σ-algebra generated by the process Yu .

Introduction

Finance and insurance companies are facing a wide range of mathematical problems. The former consider, for instance, the valuation of common derivative securities (call and put options, forward and future contracts, swaps) in complete or incomplete financial markets as well as the economic consumption and investment problem; the latter consider the tarification of insurance premia and claims reserving. The present volume is dedicated to the statistical estimation of the key parameters in some of the aforementioned considerations in finance and insurance. The notion of statistical experiment generated by an observation sample is introduced in the first part of this book with the notion of statistical inference of the unknown parameter (and the notion of sequence of estimators). In order to evaluate a sequence of estimators when the size of the observation sample increases and then to compare it with others, several asymptotic properties of sequences of estimators are defined, such as consistency, asymptotic normality and asymptotic efficiency. Different sequences of estimators of the unknown parameter (maximum likelihood estimators and Bayesian estimators) are also presented. Statistical experiments generated by a sample of independent and identically distributed random variables are relatively common in the aforementioned applications. Such classical experiments are well understood theoretically, especially those with probability measures of exponential type. They are fully described in this book, and we pay special attention to the

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Statistical Inference in Financial and Insurance Mathematics with R

so-called Gaussian shift statistical experiment. Indeed, in this setting, the sequence of maximum likelihood estimators is consistent, asymptotically normal and asymptotically efficient. However, finance and insurance applications also offer non-classical statistical experiments. Three examples of non-classical statistical experiments are treated in this book. First, the generalized linear models are studied. They extend the standard regression model to non-Gaussian distributions. In this case, the random variables of the observation sample are neither identically distributed nor Gaussian. These models are famous for the tarification of insurance premia and are described in the second part of this book. In these statistical experiments, the sequence of maximum likelihood estimators of the unknown parameter is shown to be consistent and asymptotically normal under proper assumptions. This sequence of estimators is generally not in a closed form and the computation of particular recursive numerical approximations is also presented for practical use. Second, statistical experiments implying homogeneous Markov chains are considered. They simply involve observing the price of an asset (on a regular temporal grid) considered as the solution of a specific stochastic differential equation. In this setting, the coordinates of the observation sample are neither independent nor Gaussian. In these experiments, consistent, asymptotically normal and asymptotically efficient sequences of estimators can also be built. Parametric estimation in this dependent setting is described in the third part of this book in order to calibrate the volatility parameter in pricing financial tools or calibrate short-term spot rate models. Third, various statistical experiments generated by the observation of the fractional Gaussian noise are also described. In these experiments, the random variables of the observation sample are Gaussian but dependent on a possible long-memory property. Here again, the sequence of maximum likelihood estimators is shown to be consistent, asymptotically normal and asymptotically efficient. In this book, asymptotic properties of diverse sequences of estimators are detailed. The notion of asymptotical efficiency is discussed for the different

Introduction

xv

statistical experiments considered in order to give a proper sense of estimation risk. Eighty examples illustrate the theoretical developments in this book. Moreover, computations with the R software are given throughout the text.

1 Statistical Experiments

Let Θ be an open set of p . For all n ≥ 1, the observation sample X (n) is  n the function defined by X (n) (x) = x for all x ∈ i=1 . The observation sample is possibly written as X (n) = (X1 , . . . , Xn ); each coordinate is the identity function on as well. This book will consider parametric statistical experiments generated by the observation sample X (n) and defined by the sequence  n   n   , B , (Pn,ϑ , ϑ ∈ Θ) , n ≥ 1 . [1.1] i=1

i=1

Here, for  all n ≥ 1, (Pn,ϑ , ϑ ∈ Θ) is a parametric family of probability measures on ni=1 B . E XAMPLE 1.1.– Let us toss a coin n times successively and independently. In this sequence, 0 is given for heads and 1 for tails. Possible outcomes  of this experiment are given by n-uplets x = (x1 , . . . , xn ) ∈ ni=1 {0, 1} ⊂ ni=1 . Has this coin equally likely outcomes? In order to answer the previous question, let us denote ϑ ∈ (0, 1) ⊂ the probability of obtaining 1 in a single toss. In probability  theory, the random n-uplet X (n) is considered on n n the probability space ( i=1 , i=1 B , Pn,ϑ ). Here, the probability measure Pn,ϑ has the set ni=1 {0, 1} as support and Pn,ϑ ({x}) = ϑn1 (1 − ϑ)n−n1  where n1 is the number of 1 in the n-uplet x ∈ ni=1 {0, 1}. A parametric statistical experiment as in [1.1] is described, allowing the unknown parameter

4

Statistical Inference in Financial and Insurance Mathematics with R

ϑ to belong to an open set Θ ⊂ [0, 1] ⊂ increase.

and the number of trials n ≥ 1 to

E XAMPLE 1.2.– Let us perform n measurements on a standard (etalon) successively and independently. These measurements generally present bias and variance. What is the value of this bias and this variance? In order to 2. answer the previous question, let us denote ϑ = (μ, σ 2 ) ⊂ × + ∗ ⊂ This experiment can be modeled by the observation of n possible outcomes of 2 independent Gaussian random variables of mean μ (bias) and variance n σ . The possible outcomes are given by n-uplets x = (x1 , . . . , xn ) ∈ i=1 . Here, the probability measure Pn,ϑ is defined by 

ˆ Pn,ϑ (B) =

(2πσ 2 ) B

−n 2

n 1  exp − 2 (xi − μ)2 2σ

 νn (dx)

i=1

  for all B ∈ ni=1 B and νn is the Lebesgue measure on ni=1 B . Then, a parametric statistical experiment as in [1.1] is described, allowing the 2 and the unknown parameter ϑ to belong to an open set Θ ⊂ × + ∗ ⊂ number of trials n ≥ 1 to increase. E XAMPLE 1.3.– The statistical experiment described in the previous example 1.2 can be easily generalized. Let us consider the observation of n possible outcomes of dependent Gaussian random variables of respective mean μ = (μ1 , . . . , μn ) and common covariance matrix Σ. In this setting, the random variable is neither independent nor identically distributed. The n possible outcomes are still x = (x1 , . . . , xn ) ∈ i=1 but the probability measure Pn,ϑ is defined by ˆ Pn,ϑ (B) = for all B ∈

(2π) B

n

i=1 B

−n 2

(det Σ)

− 12



1 ∗ −1 exp − (x − μ) Σ (x − μ) νn (dx) 2

.

As is described in example 1.3, statistical experiments which go beyond samples of independent and identically distributed random variables can be described by model [1.1].

Statistical Experiments

5

1.1. Dominated and homogeneous statistical experiments n Let n ≥ 1 and νn be a σ-finite measure on n i=1 B . A parametric family of probability measures (Pn,ϑ , ϑ ∈ Θ) on i=1 B is said to be dominated by νn if, for all ϑ ∈ Θ, the probability measure Pn,ϑ is absolutely continuous with respect to νn (the null sets for νn are also null sets for Pn,ϑ ). Dominated family of probability measures admits Radon–Nikodym derivatives (or probability density functions) dPn,ϑ (x) = fn (ϑ, x), dνn

x∈

n 

,

ϑ ∈ Θ,

i=1

satisfying 0 ≤ fn (ϑ, x) < ∞ for νn -almost all x ∈ all B ∈ ni=1 B ,

n

i=1

and such that, for

ˆ Pn,ϑ (B) =

B

fn (ϑ, x)νn (dx),

for all ϑ ∈ Θ. A parametric family of probability measures (Pn,ϑ , ϑ ∈ Θ) is said to be homogeneous if, for all (ϑ, ϑ ) ∈ Θ2 , the probability measure Pn,ϑ is absolutely continuous with respect to Pn,ϑ . In a homogeneous family, the null sets are identical for each probability measure. It is worth mentioning that a homogeneous family is dominated by any probability measure of the family. nA family of probability measures dominated by a σ-finite measure n νn on B with positive density functions (for ν -almost all x ∈ n i=1 i=1 ) is  n homogeneous. Indeed, in this case, for all B ∈ i=1 B , ˆ Pn,ϑ (B) =

B

ˆ

fn (ϑ, x)νn (dx)

B

fn (ϑ, x) fn (ϑ , x)νn (dx) fn (ϑ , x)

B

fn (ϑ, x) Pn,ϑ (dx) fn (ϑ , x)

= ˆ = for all (ϑ, ϑ ) ∈ Θ2 .

6

Statistical Inference in Financial and Insurance Mathematics with R

E XAMPLE 1.4.– Let Θ ⊂ and ν be the Lebesgue measure on B . Two families of probability measures dominated by ν are considered, generally denoted as (Pϑ , ϑ ∈ Θ), with Radon–Nikodym derivatives defined, for all ϑ ∈ Θ, by f (ϑ, x) =

[ϑ− 12 ,ϑ+ 12 ] (x),

x∈

,

and

1 (x − ϑ)2 f (ϑ, x) = √ exp − , 2 2π

x∈

,

[1.2]

respectively. The first family which consists of uniform probability density functions is not homogeneous.  Indeed, for ϑ ∈ Θ and 0 < h ≤ 1 such that ϑ + h ∈ Θ, the set B = ϑ + 21 , ϑ + h + 12 is fixed and Pϑ (B) = 0 but Pϑ+h (B) = h > 0. The second family which consists of positive Gaussian probability density functions is homogeneous. E XAMPLE 1.5.– Let Θ ⊂ [0, 1] ⊂ and ν be the counting measure ν = δ0 + δ1 . It is possible to consider the family of probability measures dominated by ν with Radon–Nikodym derivatives defined, for all ϑ ∈ Θ, by f (ϑ, x) = ϑx (1 − ϑ)1−x ,

x∈

.

This family consists of positive Bernoulli probability density functions and it is homogeneous. A family of probability measures (Pn,ϑ , ϑ ∈ Θ) is identifiable if the map ϑ −→ Pn,ϑ is injective. All the definitions of family of probability measures are naturally extended to statistical experiments. The statistical experiment [1.1] is said to be dominated (respectively homogeneous, identifiable) if, for all n ≥ 1, (Pn,ϑ , ϑ ∈ Θ) is dominated (respectively homogeneous, identifiable).

Statistical Experiments

7

1.2. Experiments generated by a sample of independent and identically distributed random variables Various statistical experiments will be considered in this book in insurance and finance. Among them, statistical experiments generated by a sample of independent random variables are described by 

n 

,

n 

i=1

 B , (Pn,ϑ , ϑ ∈ Θ)

 ,n≥1

i=1

with Pn,ϑ =

n  i=1

Pϑi .

 Here, for i ≥ 1, Pϑi , ϑ ∈ Θ is a family of probability measures on B dominated by a σ-finite measure ν on B with Radon–Nikodym derivatives dPϑi (x) = f i (ϑ, x), dν

x∈

,

ϑ ∈ Θ.

It can be shown that the measure ν ⊗n =

n 

ν

[1.3]

i=1

n on is also σ-finite. Moreover, the family of tensored probability i=1 B n measures (Pn,ϑ , ϑ ∈ Θ) on is dominated by νn with i=1 B Radon–Nikodym derivatives n

 dPn,ϑ f i (ϑ, xi ), (x) = dν ⊗n i=1

x = (x1 , . . . , xn ) ∈

n 

.

i=1

For instance, classical linear regression and generalized linear models enter this class of statistical experiments and will be described further.

8

Statistical Inference in Financial and Insurance Mathematics with R

A particular case is the statistical experiments generated by a sample of independent and identically distributed random variables. They are defined by [1.1] with Pn,ϑ =

n 

Pϑ

i=1

where (Pϑ , ϑ ∈ Θ) is a family of probability measures on B dominated by a σ-finite measure ν on B . It is worth mentioning that (Pn,ϑ , ϑ ∈ Θ) is consecutively dominated by ν ⊗n with Radon–Nikodym derivatives n

 dPn,ϑ (x) = f (ϑ, xi ), dν ⊗n

n 

x = (x1 , . . . , xn ) ∈

i=1

.

i=1

E XAMPLE 1.6.– Let ν be the Lebesgue measure on B and, for all ϑ ∈ Θ ⊂ , Pn,ϑ be the probability measure dominated by ν with Gaussian probability density function given by [1.2]. Let us consider an observation sample of independent and identically distributed Gaussian random variables of mean ϑ ∈ Θ ⊂ and variance 1. In this setting, the probability measure Pn,ϑ = ni=1 Pϑ admits a density function with respect to ν ⊗n given by   n n   n dPn,ϑ 1 −2 2 (x) = (2π) exp − (xi − ϑ) , x = (x1 , . . . , xn ) ∈ . dν ⊗n 2 i=1

i=1

1.3. Probability measures of exponential type Let Θ be an open set of p and ν be a σ-finite measure on B . An important class of parametric families of probability measures is the families of exponential type. A parametric family of probability measures (Pϑ , ϑ ∈ Θ) on B is of exponential type with respect to ν if it is dominated by ν and there are measurable functions h : R → R and T : R → Rp such that dPϑ (x) = C(ϑ)h(x) exp ( ϑ, T (x)) , dν

x∈

,

[1.4]

with C(ϑ) > 0 for all ϑ ∈ Θ. Here, 0 ≤ h < ∞ for ν-almost all x ∈ R and T is minimal, i.e. the following equality

ϑ, T (x) = c implies ϑ = c = 0.

for ν-almost all x ∈ R and fixed c ∈

Statistical Experiments

9

1.3.1. Properties of families of exponential type In this section, several interesting results are stated for families of probability measures of exponential type. The first two results show homogeneity and identifiability. P ROPOSITION 1.1.– A family of probability measures of exponential type is homogeneous. Proof. By definition, h(x) is non-negative and h(x) < +∞ for ν-almost all x ∈ in equation [1.4]. If ν is σ-finite and h(x) < +∞ for ν-almost all x ∈ n , then h · ν is σ-finite. Moreover, dPϑ (x) = C(ϑ) exp ( ϑ, T (x)) > 0 d(h · ν)

for h · ν-almost all x ∈

.

[1.5]

Finally, a family of probability measures dominated by a σ-finite measure μ with positive density functions (for μ-almost all x ∈ ) is homogeneous.  P ROPOSITION 1.2.– A family of probability measures of exponential type is identifiable. Proof. Let us suppose that Pϑ = Pψ . Then, for ν-almost all x ∈

,

ϑ, T (x) − ln (C(ϑ)) = ψ, T (x) − ln (C(ψ)) and

ϑ − ψ, T (x) = ln

C(ϑ) C(ψ)

.

Since T is minimal, we have that ϑ = ψ and the identifiability property.  Let us define K(ϑ) =

1 = C(ϑ)

ˆ exp ( ϑ, T (x)) h · ν(dx)

[1.6]

10

Statistical Inference in Financial and Insurance Mathematics with R

and the natural parameter set of a family of probability measures of exponential type N = {ϑ ∈

p

, K(ϑ) < ∞} .

[1.7]

will be used in The notation T (x) = (T1 (x), . . . , Tp (x))∗ ∈ p , x ∈ the following. A family of probability measures of exponential type also enjoys the following key properties. ˚ and, for T HEOREM 1.1.– The function ϑ −→ K(ϑ) is smooth at any ϑ ∈ N p any q = (q1 , . . . , qp ) ∈ , ˆ

  q1 T (x) . . . Tpqp (x) exp ( ϑ, T (x)) h · ν(dx) < ∞ 1

and ∂ q1 +...+qp K q (ϑ) = ∂ϑq11 . . . ∂ϑpp

ˆ

q

T1q1 (x) . . . Tp p (x) exp ( ϑ, T (x)) h · ν(dx).

Proof. First, the set N is convex. Indeed, the exponential function is convex and 

 exp τ ϑ + (1 − τ )ϑ , T (x)   

≤ τ |exp ( ϑ, T (x))| + (1 − τ ) exp ϑ , T (x)  for all x ∈ , all (ϑ, ϑ ) ∈ N 2 and 0 < τ < 1, which gives the convexity of N . The proof is now set in dimension one (p = 1). Let us remark that, for all y ∈ and c > 0, |y| ≤

exp(cy) + exp(−cy) . c

˚ and c sufficiently small such that ϑ ± c ∈ N , it follows: For q1 = 1, ϑ ∈ N    ∂     ∂ϑ exp (ϑT (x)) = |T (x)| exp (ϑT (x)) ≤

exp((ϑ + c)T (x)) + exp((ϑ − c)T (x)) . c

Statistical Experiments

11

Consequently, the left-hand side function is dominated by an integrable function (with respect to h · ν) and the corollary for differentiation of the dominated convergence theorem can be applied, namely ˆ ∂ ∂K(ϑ) = exp (ϑT (x)) h · ν(dx) ∂ϑ ∂ϑ ˆ = T (x) exp (ϑT (x)) h · ν(dx). For q1 > 1, dominated convergence theorem gives also the result with the use of following inequality: |y|

q1

≤

exp(cy) + exp(−cy) c

q1

≤

exp(cq1 y) + exp(−cq1 y) . c q1

˚, the constant c is chosen sufficiently small such that ϑ ± cq1 ∈ For ϑ ∈ N N.  As a corollary, it is possible to compute the first and the second moments of the random variable T in terms of the partial derivatives of the function ϕ(ϑ) = log K(ϑ) = − log C(ϑ).

[1.8]

Namely, we can show: ˚, P ROPOSITION 1.3.– For any ϑ ∈ N ˆ T (x)Pϑ (dx) = ∇ϕ(ϑ) Eϑ (T ) =

[1.9]

and, for all 1 ≤ i, j ≤ p, Covϑ (Ti , Tj ) =

∂2ϕ (ϑ). ∂ϑi ∂ϑj

[1.10]

12

Statistical Inference in Financial and Insurance Mathematics with R

Proof. The proof is set in dimension one. Theorem 1.1 gives Eϑ (T ) dK = (ϑ) C(ϑ) dϑ

and

 d2 K 1  Varϑ (T ) + (Eϑ (T ))2 = (ϑ). C(ϑ) dϑ2

Direct computations of the derivatives of ϕ give dϕ 1 dK (ϑ) = · (ϑ) and dϑ K(ϑ) dϑ

2 d2 ϕ dK d2 K 1 1 (ϑ) · (ϑ) = − · + (ϑ) dϑ2 K(ϑ)2 dϑ K(ϑ) dϑ2 

and the result.

The last result deals with statistical experiments generated by a sample of independent and identically distributed random variables. P ROPOSITION 1.4.– We suppose that (Pϑ , ϑ ∈ Θ) is of exponential type with respect to the σ-finite measure ν. Let us denote Pn,ϑ =

n 

Pϑ

i=1

the probability measure of the corresponding sample of independent and identically distributed random variables. Then, (Pn,ϑ , ϑ ∈ Θ) is of exponential type [1.4] with respect to ν ⊗n . Proof. Let dPϑ (x) = C(ϑ)h(x) exp ( ϑ, T (x)) , dν

x∈

.

The family of probability n measures (Pn,ϑ , ϑ ∈ Θ) is dominated by the σ⊗n finite measure ν on i=1 B with Radon–Nikodym derivatives   n n   dPn,ϑ n (x) = C(ϑ) h(xi ) exp ϑ, T (xi ) , dν ⊗n i=1

x = (x1 , . . . , xn ) ∈

i=1

n  i=1

R,

Statistical Experiments

13

which are of exponential type [1.4] for all ϑ ∈ Θ. For minimality, we suppose that

ϑ,

n 

T (xi ) = c

for ν-almost all

i=1

x = (x1 , . . . , xn ) ∈

n 

R and fixed c ∈

.

i=1

Taking vectors (x1 , a, . . . , a) with fixed a ∈

,

ϑ, T (x1 ) = c − (n − 1) ϑ, T (a) is obtained and ϑ = c = 0 by minimality of T .



Consequently, statistical experiments generated by a sample of independent and identically distributed random variables with families of probability measures of exponential type are homogeneous and identifiable. 1.3.2. Classic examples of families of exponential type In this section, several examples of families of probability measures of exponential type are given. 1.3.2.1. Densities with respect to the Lebesgue measure Let ν be the Lebesgue measure on B which is σ-finite. E XAMPLE 1.7.– The probability density function (with respect to ν) of a Gaussian variable of mean μ and standard deviation σ can be written as

(x − μ)2 1 [1.11] exp − f (x) = √ , x∈ . 2σ 2 2πσ 2 1) Let us consider that the variance σ 2 is known. A parametric family of probability measures (Pϑ , θ ∈ Θ) can be set with ϑ = σμ2 , Θ = (a, b) ⊂ N = and probability density functions 2 2

 1

dPϑ ϑ σ x2 2 −2 (x) = 2πσ exp − exp − 2 exp (ϑx) , x ∈ . dν 2 2σ

14

Statistical Inference in Financial and Insurance Mathematics with R

 2 2 − 1

For instance, quantities C(ϑ) = 2πσ 2 2 exp − ϑ 2σ , h(x) =   x2 exp − 2σ and T (x) = x can be identified in order to show the exponential 2 type of the probability density function. Direct computations or application of proposition 1.3 lead to Eϑ (T ) = ϑσ 2

and Varϑ (T ) = σ 2 .

2) Let us consider that the mean μ is known. A parametric family of probability measures can be set with ϑ = σ12 , Θ = (a, b) ⊂ N = + ∗ and probability density functions dPϑ (x) = dν



ϑ 2π

1 2



2μx − x2 ϑμ2 exp ϑ · , exp − 2 2

For instance, quantities C(ϑ) = 2

ϑ 2π

 12

x∈

.

  2 exp − ϑμ2 , h(x) = 1 and

T (x) = μx − x2 can be identified. Direct computations or application of proposition 1.3 lead to Eϑ (T ) =

1 μ2 − 2 2ϑ

and

Varϑ (T ) =

1 . 2ϑ2

3) In the general setting, a parametric family of probability measures can be set with p = 2, ϑ1 = σμ2 , ϑ2 = σ12 , Θ = (a, b) × (c, d) ⊂ N = × + ∗ and probability density functions dPϑ (x) = dν



ϑ2 2π

1 2



ϑ21 exp − exp ( ϑ, T (x)) , 2ϑ2 

Here, quantities C(ϑ) = T (x) =

x 2 − x2



ϑ2 2π

1 2

x∈

  ϑ2 exp − 2ϑ12 , h(x) = 1 and

.

Statistical Experiments

15

can be identified. Direct computations or application of proposition 1.3 lead to ⎞ ⎛ ⎞ ⎛ ϑ1 ϑ1 1 − 2 ϑ2  

ϑ2 ϑ2 ⎠. 2 ⎠ and Cov(T ) = ⎝ Eϑ (T ) = ⎝ 1 1 ϑ2 − 2 ϑ2 + ϑϑ12 − ϑ12 1 2 + 13 ϑ2 2ϑ2

ϑ2

E XAMPLE 1.8.– The probability density function of an exponential variable of rate β can be written as dPϑ (x) = β exp (−βx) dν

+

(x),

x∈

.

A parametric family of probability measures can be set with ϑ = −β and Θ = (a, b) ⊂ N = − . The quantities C(θ) = −ϑ, h(x) = + (x) and T (x) = x can be set. Direct computations or application of proposition 1.3 lead to Eϑ (T ) = −

1 ϑ

and

Varϑ (T ) =

1 . ϑ2

E XAMPLE 1.9.– The probability density function of a Weibull variable of shape parameter k ≥ 1 and scale parameter ρ is given by k dPϑ (x) = dν ρ

  k−1 x x k exp − ρ ρ

+

(x),

x∈

.

If k = 1, it reduces to the exponential distribution of rate ρ1 . If k is known, this distribution enters into the family of probability measures of exponential type. Namely, ϑ = − ρ1k , Θ = (a, b) ⊂ N = − , C(ϑ) = −ϑ, h(x) =

kxk−1 + (x) and T (x) = xk + (x) can be fixed. Direct computations or application of proposition 1.3 lead to Eϑ (T ) = −

1 ϑ

and Varϑ (T ) =

1 . ϑ2

Beta, gamma and log-normal distributions are also of exponential type. The corresponding statistical experiments can be numerically simulated with the R software [RCO 17]. The command rnorm (respectively rexp, rlnorm, rgamma, rbeta, rweibull) can simulate the random variables of

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Statistical Inference in Financial and Insurance Mathematics with R

Gaussian distribution (respectively exponential, log-normal, gamma, beta, Weibull). It is also possible to access the probability density function of each distribution with the instruction dnorm (respectively dexp, dlnorm, dgamma, dbeta, dweibull). 1.3.2.2. Densities with respect to a counting measure In this section, the dominating σ-finite measure is a counting measure. It is elicited in each example for clarity. E XAMPLE 1.10.– For a binomial variable,  the probability density function with respect to the counting measure ν = m i=0 δi is given by

m x dPϑ p (1 − p)m−x (x) = x dν





m p (1 − p)m exp log x , x∈ . = x 1−p  Here, ϑ = log

p 1−p

 , Θ = (a, b) ⊂ N =

+, ∗

m x ,

h(x) =

1 C(ϑ) = (1+exp(ϑ)) m and T (x) = x can be identified. Direct computations or application of proposition 1.3 lead to

Eϑ (T ) = m Varϑ (T ) = m

exp(ϑ) = mp and 1 + exp(ϑ) exp(ϑ) = mp(1 − p). (1 + exp(ϑ))2

E XAMPLE 1.11.– For a Poisson variable,  the probability density function with respect to the counting measure ν = m≥0 δm is given by μx 1 dPϑ (x) = exp (−μ) = exp (−μ) exp (log(μ)x) , dν x! x!

x∈

.

1 + , h(x) = For instance, ϑ = ln(μ), Θ = (a, b) ⊂ ∗ x! , C(ϑ) = exp(− exp(ϑ)) and T (x) = x can be fixed. Direct computations or application of proposition 1.3 lead to

Eϑ (T ) = exp(ϑ) = μ and

Varϑ (T ) = exp(ϑ) = μ.

Statistical Experiments

17

Discrete random variables can also be simulated with R. For instance, the instructions rbinom and rpois can be used for simulating a binomial random variable and a Poisson random variable, respectively. In order to compute the corresponding probability density functions, the user can refer to the commands dbinom and dpois, respectively.

2 Statistical Inference Let Θ be an open subset of p . The statistical experiment generated by the sample X (n) (x) = x, for all x ∈ ni=1 , and defined by  n   n   , B , (Pn,ϑ , ϑ ∈ Θ) , n ≥ 1 i=1

i=1

is considered. Here, for n n≥ 1, (Pn,ϑ , ϑ ∈ Θ) is a parametric family of probability measures on dominated by a σ-finite measure νn on i=1 B n i=1 B . The problem of statistical inference in the parametric setting is to build tools (and evaluate these tools)in order to estimate the unknown parameter ϑ ∈ Θ observing the value x ∈ ni=1 of the sample X (n) . The reader can refer to examples 1.1 and 1.2 in section 1 where two simple estimation problems are discussed. These tools are called sequences of estimators later on and some of their properties are studied, namely bias, variance, consistency, asymptotic normality and asymptotic efficiency. In the following, a sequence ofstatistics or sequence of estimators (Tn , n ≥ 1) is a sequence of ( ni=1 B , pi=1 B ) - measurable maps Tn : n −→ p , for all n ≥ 1. 2.1. Asymptotic properties of sequences of estimators Asymptotic properties of several sequences of statistics (maximum likelihood estimators, Bayesian estimators, etc.) are discussed in this section for different parametric statistical experiments.

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Statistical Inference in Financial and Insurance Mathematics with R

2.1.1. Consistency Let ϑ ∈ Θ. A sequence of statistics (Tn , n ≥ 1) is a consistent sequence of estimators if, for all ε > 0, lim Pn,ϑ (|Tn − ϑ| > ε) = 0.

[2.1]

n→∞

Here, |·| is the usual norm on p . In other words, a consistent sequence of estimators converges in probability to the true parameter when the sample size tends to infinity; this can be also noted Tn −→ ϑ in probability as n −→ ∞. The existence of a consistent sequence of estimators for statistical experiments generated by a sample of independent and identically distributed random variables has been set under very general assumptions (see [IBR 81, theorem 4.1, p. 32]). This result does not give any construction method for such sequences of estimators. Several examples of consistent sequences of estimators in statistical experiments generated by independent and identically distributed samples are given below. E XAMPLE 2.1.– Let ν be the Lebesgue measure on B . Let us consider the example 1.6 of estimating the mean ϑ in a sample of independent and identically distributed Gaussian variables with variance 1. In this statistical experiment, for all ϑ ∈ Θ = (a, b) ⊂ ,   n  n dPn,ϑ 1 (x) = (2π)− 2 exp − (xi − ϑ)2 , dν ⊗n 2 i=1

x = (x1 , . . . , xn ) ∈

n 

.

i=1

This statistical experiment is called the Gaussian shift experiment. Since Gaussian distribution falls into the family of probability measures of the exponential type, it is worth mentioning that the statistical experiment is identifiable. The sequence of empirical means (Tn , n ≥ 1) is defined with n

Tn (x) =

1 xi , n i=1

x = (x1 , . . . , xn ) ∈

n  i=1

.

Statistical Inference

21

In this setting, the law (under Pn,ϑ ) of the statistical error Tn − ϑ is the Gaussian distribution of mean zero and variance n1 . Consecutively, the consistency for this sequence of estimators can be proved with Markov’s inequality:

Eϑ (Tn − ϑ)2 1 Pn,ϑ (|Tn − ϑ| > ε) ≤ = 2. 2 ε nε The consistency of the sequence of estimators can be checked numerically with the following list of R commands: theta