Elliptically Contoured Models in Statistics

Softcover Nachdruck of the original 1. Auflage 1993
ISBN: 978-94-010-4719-7
Verlag: Springer Netherlands

Buch, Englisch, Band 240, 327 Seiten, Paperback, Format (B × H): 160 mm x 240 mm, Gewicht: 545 g

Reihe: Mathematics and Its Applications

In multivariate statistical analysis, elliptical distributions have recently provided an alternative to the normal model. Most of the work, however, is spread out in journals throughout the world and is not easily accessible to the investigators. Fang, Kotz, and Ng presented a systematic study of multivariate elliptical distributions, however, they did not discuss the matrix variate case. Recently Fang and Zhang have summarized the results of generalized multivariate analysis which include vector as well as the matrix variate distributions. On the other hand, Fang and Anderson collected research papers on matrix variate elliptical distributions, many of them published for the first time in English. They published very rich material on the topic, but the results are given in paper form which does not provide a unified treatment of the theory. Therefore, it seemed appropriate to collect the most important results on the theory of matrix variate elliptically contoured distributions available in the literature and organize them in a unified manner that can serve as an introduction to the subject. The book will be useful for researchers, teachers, and graduate students in statistics and related fields whose interests involve multivariate statistical analysis. Parts of this book were presented by Arjun K Gupta as a one semester course at Bowling Green State University. Some new results have also been included which generalize the results in Fang and Zhang. Knowledge of matrix algebra and statistics at the level of Anderson is assumed. However, Chapter 1 summarizes some results of matrix algebra.

Zielgruppe

Research

Autoren/Hrsg.

Varga, Tamas

Gupta, Arjun K.

Fachgebiete

• Wirtschaftswissenschaften Wirtschaftswissenschaften Wirtschaftswissenschaften: Allgemeines
• Mathematik | Informatik Mathematik Stochastik
• Mathematik | Informatik Mathematik Algebra

Inhaltsverzeichnis

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.