Multivariate Statistics: Theory And Applications - Proceedings Of The Ix Tartu Conference On Multivariate Statistics And Xx International Workshop On Matrices And Statistics

Tõnu Kollo

• Publisher: WSPC
• Year: 2013
• Print ISBN: 9789814449397
• eBook ISBN: 9789814449410
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

The book aims to present a wide range of the newest results on multivariate statistical models, distribution theory and applications of multivariate statistical methods. A paper on Pearson–Kotz–Dirichlet distributions by Professor N Balakrishnan contains main results of the Samuel Kotz Memorial Lecture. Extensions of linear models to multivariate exponential dispersion models and Growth Curve models are presented, and several papers on classification methods are included. Applications range from insurance mathematics to medical and industrial statistics and sampling algorithms.

Contents:

• Variable Selection and Post-Estimation of Regression Parameters Using Quasi-Likelihood Approach (S Fallahpour and S E Ahmed)
• Maximum Likelihood Estimates for Markov-Additive Processes of Arrivals by Aggregated Data (A M Andronov)
• A Simple and Efficient Method of Estimation of the Parameters of a Bivariate Birnbaum-Saunders Distribution Based on Type-II Censored Samples (N Balakrishnan and X Zhu)
• Analysis of Contingent Valuation Data with Self-Selected Rounded WTP-Intervals Collected by Two-Steps Sampling Plans (Yu K Belyaev and B Kriström)
• Optimal Classification of Multivariate GRF Observations (K Dučinskas and L Dreižienė)
• Multivariate Exponential Dispersion Models (B Jørgensen and J R Martínez)
• Statistical Inference with the Limited Expected Value Function (M Käärik and H Kadarik)
• Shrinkage Estimation via Penalized Least Squares in Linear Regression with an Application to Hip Fracture Treatment Costs (A Liski, E P Liski and U Häkkinen)
• K-Nearest Neighbors as Pricing Tool in Insurance: A Comparative Study (K Pärna, R Kangro, A Kaasik and M Möls)
• Statistical Study of Factors Affecting Knee Joint Space and Osteophytes in the Population with Early Knee Osteoarthritis (T von Rosen, A E Tamm, A O Tamm and I Traat)
• Simultaneous Confidence Region for ρ and σ 2 in a Multivariate Linear Model with Uniform Correlation Structure (I Žežula and D Klein)

Readership: Graduated students and Professional researchers in mathematics.

MULTIVARIATE EXPONENTIAL DISPERSION MODELS

B. J∅RGENSEN

Department of Mathematics and Computer Science,

University of Southern Denmark, Campusvej 55,5230 Odense M, Denmark

E-mail: [email protected]

J. R. MARTÍNEZ

FAMAF, Universidad Nacional de Cόordoba

Ciudad Universitaria, 5000 Cόordoba,Argentina

E-mail: [email protected]

We develop a new class of multivariate exponential dispersion models with a fully flex_ible correlation structure, and present multivariate versions of the Poisson, binomial, negative binomial, gamma, inverse Gaussian, and Tweedie distributions, some of which extend ex_isting bivariate or multivariate models of these types. The new models are constructed using an extended convolution method, which interpolates between the set of fully correlated pairs of variables and the set of independent marginals of the prescribed form. The models have an additive as well as a reproductive form. The additive form is particularly suitable for discrete data, whereas the reproductive form is more suitable for continuous data, having properties similar to the multivariate normal distribution. Multivariate exponential dispersion models are useful as error distributions for multivariate generalized linear models for the purpose of modelling multivariate non-normal data.

Keywords: Convolution method; Multivariate binomial distribution; Multivariate dispersion model; Multivariate gamma distribution; Multivariate generalized linear model; Multivariate inverse Gaussian distribution; Multivariate negative binomial distribution; Multivariate Poisson distribution; Multivariate Tweedie distribution.

1. Introduction

The aim of this investigation is to construct multivariate distributions with margins of prescribed form, having a fully flex_ible correlation structure. Our motivation is to be able to model multivariate non-normal data by some form of multivariate generalized linear model, which in turn requires a suitable form of multivariate exponential dispersion model. Such multi variate models were introduced by Jørgensen,¹ who considered the general problem of constructing multivariate dispersion models, extending the univariate case studied in detail by Jørgensen.²

There is no shortage of multivariate non-normal distributions available, but the class of multivariate exponential dispersion models proposed by Jørgensen1 has the advantage of possessing a fully flex_ible correlation structure, while having margins of the prescribed form. A multivariate exponential dispersion model in its reproductive form (cf. Sec. 5) is parametrized by a k-vector of means μ and a symmetric positive-definite k×k dispersion matrix Σ, thereby resembling the multivariate normal distribution in much the same way that univariate exponential dispersion models resemble the univariate normal distribution.²

Multivariate exponential dispersion models of additive form (cf. Sec. 3) are constructed by an extended convolution method, which interpolates between the set of fully correlated pairs of variables and the set of independent margins of the prescribed form. This method explores the convolution property of conventional additive exponential dispersion models (cf. Sec. 2) in order to generate the desired number of parameters, namely k means and k(k + 1)/2 variance and covariance parameters. Multivariate additive exponential dispe...

Table of contents

• Cover 
• Half Title
• Title
• Copyright
• Preface
• Organizing Committees
• Contents
• Variable Selection and Post-Estimation of Regression Parameters Using Quasi-Likelihood Approach
• Maximum Likelihood Estimates for Markov-Additive Processes of Arrivals by Aggregated Data
• A Simple and Efficient Method of Estimation of the Parameters of a Bivariate Birnbaum-Saunders Distribution Based on Type-II Censored Samples
• Analysis of Contingent Valuation Data with Self-Selected Rounded WTP-Intervals Collected by Two-Steps Sampling Plans
• Optimal Classification of Multivariate GRF Observations
• Multivariate Exponential Dispersion Models
• Statistical Inference with the Limited Expected Value Function
• Shrinkage Estimation via Penalized Least Squares in Linear Regression with an Application to Hip Fracture Treatment Costs
• K-Nearest Neighbors as Pricing Tool in Insurance: A Comparative Study
• Statistical Study of Factors Affecting Knee Joint Space and Osteophytes in the Population with Early Knee Osteoarthritis
• Simultaneous Confidence Region for ρ and σ2 in a Multivariate Linear Model with Uniform Correlation Structure
• Index