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Multivariate calculation. Use of the continuous groups. (English) Zbl 0561.62048
Springer Series in Statistics. New York etc.: Springer-Verlag. XVI, 376 p. DM 138.00 (1985).

This book contains a selection of mathematical topics needed for multivariate statistical inference, in particular for the solution of sampling distributions problems which lead to multivariate normal density functions.

After a very useful chapter of introduction and survey, there are 13 chapters headed as follows: Transforms (2); Locally compact groups and Haar measure (3); Wishart’s paper, dealing with the paper Biometrika 20, 32-52 (1928) (4); The Fubini-type theorems of Karlin (5); Manifolds and exterior differential forms (6); Invariant measures on manifolds (7); Matrices, operators, null sets (8); Examples using differential forms (9); Cross-sections and maximal invariants (10); Random variable techniques (11); Zonal polynomials (12,13), and Multivariate inequalities (14). About this contents the author comments that ”Chapters 2,4,5,9,10 and 11 are directly concerned with distinct techniques of computing or otherwise determining density functions, while Chapters 3,6,7 and 8 give the development of needed mathematical background”.

This book is an extension, revision and correction of another one by the same author, Techniques of multivariate calculation. (1976; Zbl 0337.62033). The level of the present book is advanced, and assumes a great deal of its readers. This includes measure theory, measures in metric spaces, locally compact groups and Hausdorff spaces, plus other standard techniques like quadratic forms, positive definite matrices and canonical forms.

The book is intended mainly to be used by researchers and advanced students interested in the mathematical aspects of some parts of multivariate statistical analysis. For this audience the material is excellent, the presentation aims at completeness, and references to the main sources are frequent and detailed. Some references to statistical problems (analysis of variance or canonical correlations) are given; however this is not a book on statistics but on mathematical methods which are potentially useful for some statistical problems.

In conclusion, this is a mathematically advanced treatment of approaches, methods and techniques to be used by mathematical statisticians working in some important areas of multivariate inference. The presentation of the material (and of the book) is very good, and for the indicated audience the availability of a single source containing this material will no doubt be a great asset.

Reviewer: R.Mentz
MSC:
62H05Characterization and structure theory (Multivariate analysis)
62-02Research monographs (statistics)
28C10Set functions and measures on topological groups or semigroups, etc.
62H10Multivariate distributions of statistics
62A01Foundations and philosophical topics in statistics
58A15Exterior differential systems (Cartan theory)