Elliptically contoured models in statistics. (English) Zbl 0789.62037

Mathematics and its Applications (Dordrecht) 240. Dordrecht: Kluwer Academic Publishers (ISBN 0-7923-2115-4/hbk). x, 327 p. (1993).
The theory of elliptically contoured distributions has been rapidly developed in the past twenty five years. Recently, there are some books published on the theory of elliptically contoured distributions and their applications in multivariate analysis; for example, K.-T. Fang, S. Kotz and K.-W. Ng [Symmetric multivariate and related distributions. (1990; Zbl 0699.62048)], K.-T. Fang and Y.-T. Zhang [Generalized multivariate analysis. (1990; Zbl 0724.62054)], and K.-T. Fang and T. W. Anderson (eds.) [Statistical inference in elliptically contoured and related distributions. (1990)]. The present book gives a systematic review of the matrix variates elliptically contoured distributions (MECD) and their applications in statistical inference. Many results are taken from the three books just mentioned, but some new results that have not been published in any journal are also included.
The book includes nine chapters. Chapter 1 summarizes notations, some results of matrix algebra, and the related literature. Chapter 2 introduces basic properties of MECD, such as the density, marginal distribution, conditional distribution, stochastic representation, mean and covariance. More detailed discussion on density functions and expected values is given in Chapter 3. The authors present various moments of MECD, their quadratic forms and related functions. Many of these results are obtained by the authors. In Chapter 4, the stochastic representation and the inverse Laplace transform are used to treat mixtures of normal distributions which are the most important subclass of MECD. The distribution and rank of quadratic forms and related functions are given in Chapter 5. An extension of Cochran’s theorem for MECD is also discussed. Characterizations based on invariance and characterizations of normality are given in Chapter 6. The last three chapters are devoted to statistical inference in MECD: estimation, hypotheses testing, and linear models giving extensions of the theory of statistical inference in multivariate normal populations to this general family.
The book is well written. The authors present detailed proofs for a number of theorems so that the reader can easily follow. The authors collected many important references in their book. It seems that the authors do not make effort to collect as many as possible up-to-date references. For example, there are a number of nice results on moments of elliptically contoured distributions by D. V. Rosen, C. S. Wong and Steyn that do not appear in this book. There are several families of matrix variates elliptically contoured distributions (cf. Fang and Zhang, 1990), the most results in this book are concerned with the smallest family only.


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