This is a book on multivariate statistical analysis, written at a good level of mathematical completeness, with emphasis on methods based on the multivariate normal distribution. In the preface the authors express: ”In this book the maximum likelihood method and Roy’s union-intersection principle are employed to obtain test statistics and confidence [regions]. Emphasis is given to asymptotic approximations to the distributions of test statistics as well as estimators and their use in the analysis. Usually nonnull distributions (or power functions) are quite complicated, even in asymptotic expansion, and hence these materials are omitted, except for some particular results needed in the discussion... Also emphasis is on multiple comparisons among populations and among procedures.”
There are ten chapters on standard topics: multivariate normal distribution, Wishart distribution, regression and correlation, inference on mean vector, multivariate regression and linear model, inference on covariance matrices, discriminant analysis, characteristic roots and vectors, principal components, and canonical correlations. Chapter 6 deals with multiple comparisons on mean vectors, chapter 4 has a useful treatment of asymptotic expansion formulas, and chapter 13 collects several topics on the selection of variables.
There are appendices on matrices and matrix derivatives and Jacobians, written in rather abridged form, but with very useful material. A final appendix contains Fortran listings of five computer programs for multivariate calculations: the first one deals with tests of multivariate normality, and it may be of use since this topic is not commonly encountered in standard computer programs or packages; the other four deal with linear discrimination (2 programs), principal components and canonical correlations.
The topics covered may be characterized as traditional. Robustness considerations are discussed in some detail only for Hotelling’s \(T^ 2\), and Bayesian arguments enter explicitly only when dealing with discrimination procedures. Brief numerical illustrations are provided in various parts of the text, but no detailed data analysis is attempted or suggested; this is consistent with the approach of the book, and the student is expected to have developed some feeling for the analysis of multivariate observations through a methods course taken before studying the present text (authors’ preface).
The approach in the book is quite formal, with numbered definitions, theorems, corollaries, and notes. These leave little space for other types of comments. Each chapter is followed by a collection of exercises, for which no solutions are provided in the text. There is an extensive list of references (40 pages) but no statistical tables.
The approach seemed very sound and reasonable to this reviewer, and the treatment carefully guided to a good understanding on the part of the reader or student. I am currently teaching an applied graduate-level course in multivariate statistical analysis, and found use for material in several sections of the book.
About the computer programs in the book, I copied and run one on linear discrimination for two groups and found the output quite simple and somewhat less detailed than the packaged program we normally use. In fact, there is at least one program in which a warning is inserted about it being incomplete. In view of the approach of the book, and of the nature of the programs incorporated in this appendix (taking 70 pages), this may be the part of the book that is most difficult to justify.
In summary, this reviewer liked this book, and found that several parts of it contained good material to use in a course or as reference material. A course based on this book will be demanding on the mathematical and statistical background of the studends, and this has been the purpose of the authors in writing it. Recently 16 graduate-level books on multivariate analysis were reviewed [see P. K. Sen, Contemporary textbooks on multivariate statistical analysis: a panoramic appraisal and critique. J. Am. Stat. Assoc. 81, 560-564 (1986)], and I would rate this book by Siotani and his colleagues as a strong competitor of those oriented to a traditional, theoretical, and mathematically sound treatment of the subject.