Elliptically Contoured Models in Statistics - Arjun K. Gupta

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Résumé :
This volume presents the first detailed introduction to the theory of matrix variate elliptically contoured distributions.br/ The book comprises eight chapters and an up-to-date bibliography. Chapter 1 summarizes some results of matrix algebra. Chapter 2 deals with the basic properties of matrix variate elliptically contoured distributions, such as the probability density function and expected values. It also presents one of the most important tools of the theory of elliptical distributions, the stochastic representation. The probability density function and expected values are investigated in detail in Chapter 3. Chapter 4 focuses on elliptically contoured distributions that can be represented as mixtures of normal distributions. The distributions of functions of random matrices with elliptically contoured distributions are discussed in Chapter 5, with special attention being paid to quadratic forms. Characterization results are given in Chapter 6. Chapters 7-9 are devoted to statistical inference.br/ For researchers and graduate students in statistics and related fields whose interests involve multivariate statistical analysis. br/

Biographie:
Arjun Gupta is affiliated with the Department of Statistics at Bowling Green State University. Tamas Varga is a statistical researcher in Hungary. Taras Bodnar is affiliated with the Department of Statistics at Stiftung European University Viadrina....

Sommaire:
1. Preliminaries.- 1.1 Introduction and Literature Review.- 1.2 Notations.- 1.3 Some Results from Matrix Algebra.- 1.4 A Functional Equation.- 2. Basic Properties.- 2.1 Definition.- 2.2 Probability Density Function.- 2.3 Marginal Distributions.- 2.4 Expected Value and Covariance.- 2.5 Stochastic Representation.- 2.6 Conditional Distributions.- 2.7 Examples.- 3. Probability Density Function and Expectedvalues.- 3.1 Probability Density Function.- 3.2 More on Expected Values.- 4. Mixture of Normal Distributions.- 4.1 Mixture by Distribution Functions.- 4.2 Mixture by Weighting Functions.- 5. Quadratic Forms and other Functions of Elliptically Contoured Matrices.- 5.1 Cochran' s Theorem.- 5.2 Rank of Quadratic Forms.- 5.3 Distribution of Invariant Matrix Variate Functions.- 6. Characterization Results.- 6.1 Characterizations Based on Invariance.- 6.2 Characterizations of Normality.- 7. Estimation.- 7.1 Maximum Likelihood Estimators of the Parameters.- 7.2 Properties of the Estimators.- 8. Hypothesis Testing.- 8.1 General Results.- 8.2 Two Models.- 8.3 Testing Criteria.- 9. Linear Models.- 9.1 Estimation of the Parameters in the Multivariate Linear Regression Model.- 9.2 Hypothesis Testing in the Multivariate Linear Regression Model.- 9.3 Inference in the Random Effects Model.- References.- Author Index.