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Matrix variate distributions. (English) Zbl 0935.62064

Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. 104. Boca Raton, FL: CRC Press. 367 p. (2000).
Random matrices appear in the theory and in applications of multivariate analysis. A common-occurring example is the \(p\times n\) “sample observation matrix”, in which each of its \(n\) columns is an independent sample from a given \(p\)-variate distribution. Similar matrices, without the conditions of independence, arise in multivariate time series, stochastic processes and repeated measurements on multivariate variables, for example.
This book deals with the probability distributions of random matrices. The work is intended to supplement the four volume treatise by N. L. Johnson and S. Kotz, Distributions in Statistics. (see the reviews Zbl 0248.62021 and Zbl 0292.62009). As such, it contains a wealth of material, some of which is new. The stated purpose is “to present most of the developments that have taken place in continuous matrix variate distribution theory in a systematic and integrated form.”
There is a review of mathematical prerequisites in Chapter 1. The next five chapters deal with the following matrix distributions: normal, Wishart, \(t\), beta, and Dirichlet. The final three chapters deal with distributions of quadratic forms, miscellaneous distributions, and general families of matrix variate distributions, respectively. In general, the format is of short comments, definition, theorem, proof, corollary. There is also a glossary of notations and abbreviations, which is very useful for this kind of book, and a 20-page list of references, “which contains only items cited in the text; it is not exhaustive, especially as regards to papers on applied topics.” A very short reference to applications is provided in the first section of Chapter 1.
In the preface the authors state that the book will be especially useful to graduate students, teachers and researchers interested in multivariate statistical analysis. One of the authors presented parts of this book in a one semester course. Every chapter is followed with a set of problems, of which there is a total of 246 in the nine chapters. The book can also serve as a source of supplementary reading and reference to many researchers. It is assumed that the reader is familiar with introductory multivariate analysis and matrix algebra.

MSC:

62H10 Multivariate distribution of statistics
62-02 Research exposition (monographs, survey articles) pertaining to statistics
15B52 Random matrices (algebraic aspects)
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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