Theory of multivariate statistics. (English) Zbl 0930.62054

Bilodeau, Martin; Brenner, David

Springer Texts in Statistics. New York, NY: Springer. xiv, 288 p. (1999).

The objective of this book is to present the main results of the modern theory of multivariate statistics to an audience of advanced students who would appreciate a concise and mathematically rigorous treatment of that material. It is intended for use as a textbook by students taking a first graduate course in the subject, as well as for a general reference of interested research workers who will find, in a readable form, developments from recently published work on certain broad topics not otherwise easily accessible, as, for instance, robust inference (using adjusted likelihood ratio tests (LRT)) and the use of the bootstrap in a multivariate setting. The references contains over 150 entries post - 1982. The main development of the text is supplemented by over 135 problems, most of which are original from the authors.
As a textbook, the first eight chapters should provide a more than adequate amount of material for coverage in one semester. These eight chapters, proceeding from a thorough discussion of the normal distribution and multivariate sampling in general, deal with random matrices, Wishart distributions, and Hotelling’s \(T^2,\) to culminate in the standard theory of estimation and testing of means and variances. The remaining six chapters treat more specialized topics than it might perhaps be wise to attempt in a simple introduction, but would easily be accessible to those already versed in the basics. Detailed chapters on multivariate regression, principal components, and canonical correlations are included, each of which should be of interest to anyone pursuing further study. The last three chapters, dealing, in turn, with asymptotic expansions, robustness and bootstrap, discuss concepts that are of current interest for active research and take the reader (gently) into territory not altogether perfectly charted. This should serve to draw one (gracefully) into the literature. The treatment is in most ways thoroughly orthodox, but in several ways novel and unique. The readers will find the approach refreshing, and perhaps even a bit liberating, particularly those saturated in a lifetime of matrix derivatives and Jacobians.
From the Contents: 1. Linear algebra. Vectors and matrices; Image space and kernels; Nonsingular matrices and determinants; Eigenvalues and eigenvectors; Orthogonal projections; Matrix decompositions. 2. Random vectors. Distribution functions; Equals-in-distribution; Discrete distributions; Expected values; Mean and variance; Characteristic functions; Absolutely continuous distributions; Uniform distributions; Joint and marginals; Independence; Change of variables; Jacobians. 3. Gamma, Dirichlet, and \(F\) distributions. 4. Invariance. Reflection symmetry; Univariate normal and related distributions; Permutation invariance; Orthogonal invariance. 5. Multivariate normal. Nonsingular, singular, conditional normal distributions; Elementary applications. Sampling the univariate normal; Linear estimation; Simple correlations; Multivariate sampling. 6. Random matrices and multivariate samples. Asymptotic distributions. 7. Wishart distributions. Joint distribution of \(\bar\mathbf x\) and \(\mathbf S\); Box-Cox transformations.
8. Tests on mean and variance. Hotelling-\(T^2\); Simultaneous confidence intervals on means; Linear hypotheses; Nonlinear hypotheses; Multiple correlations; Asymptotic moments; Partial correlations; Tests of sphericity; Tests of equality of variances; Asymptotic distributions of eigenvalues; The one-sample problem; The two-sample problem; The case of multiple eigenvalues. 9. Multivariate regression. Estimation; The general linear hypothesis; Canonical forms; LRT for the canonical problem; Invariant tests; Random design matrices \(\mathbf X\); Predictions; One-way classification. 10. Principal components. Best approximating subspaces; Sample principal components from \(\mathbf S\); Sample principal components from \(\mathbf R\); A test for multivariate normality. 11. Canonical correlations. Tests of independence; Properties of \(\mathbf U\) distributions; Q-Q plots of squared radii; Asymptotic distributions. 12. Asymptotic expansions. General expansions. 13. Robustness. Elliptical distributions; Maximum likelihood estimates (MLE); Normal MLE; Elliptical MLE; Robust estimates; \(\mathbf M\) estimates; \(\mathbf S\) estimates; Robust Hotelling-\(T^2\); Robust tests on scale matrices; Adjusted LRT; Weighted Nagao’s test for a given variance; Relative efficiency of adjusted LRT. 14. Bootstrap confidence regions and tests. Confidence regions and tests for the mean; Confidence regions for the variance; Tests on the variance.
A. Inversion formulas. B. Multivariate cumulants. Applications to asymptotic distributions. C. S-plus functions. References. Author Index. Subject Index.

Reviewer: E.Psyadlo (Odessa)

MSC:

62Hxx	Multivariate analysis
62-02	Research exposition (monographs, survey articles) pertaining to statistics
62-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics

Keywords:

multivariate sampling; random matrices; Wishart distributions; Hotelling’s T-square; testing of means and variances; multivariate regression; likelihood ratio tests; principal components