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Invariant measures on groups and their use in statistics. (English) Zbl 0803.62001
Institute of Mathematical Statistics Lecture Notes - Monograph Series 14. Hayward, CA: Institute of Mathematical Statistics (ISBN 0-940600-19-6/pbk). viii, 238 p., open access (1990).
“Groups are wonderful mathematical structures that find applications in many different fields, including statistics, as exemplified, for instance, by such recent publications as P. Diaconis, Group representations in probability and statistics. IMS Lect. Notes – Monograph Ser. 11 (1988; Zbl 0695.60012) and M. L. Eaton, Group invariance applications in statistics. Reg. Conf. Ser. Probab. Stat. 1 (1989; Zbl 0749.62005). I fell in love with groups while still a physics student and have been fascinated by their beauty and usefulness ever since.” (from the Preface.) The author’s fascination can be felt by the readers even when glancing through the book.
“This monograph deals with problems concerning distributions in statistical models in which there is a group of invariance transformations. The methods to be presented make use of mathematical tools that involve interplay between groups and integration. The purpose of this monograph is not only to demonstrate by examples the statistical usefulness of the methods, but also to present a systematic account of the mathematical background.” (from the Introduction.)
Apart from the introduction the starting two thirds of the book are devoted to the theoretical background: 2. Spaces, functions, and groups acting on spaces; 3. Differential manifolds, tangent spaces, and vector fields; 4. Differential forms on manifolds; 5. Lie groups and Lie algebras; 6. Integration on locally compact spaces according to Bourbaki; 7. Invariant and relatively invariant measures on locally compact groups and spaces. The presentation of this material is influenced by the Bourbaki school.
The second part deals with the “Factorization of measures on locally compact spaces induced by the action of a group, with help of a global cross section”. After the general theory is developed, applications to different types of problems are indicated, in particular, to MANOVA settings. The method presented is compared with that of S. A. Andersson, H. K. Brøns and S. T. Jensen [Ann. Stat. 11, 392-415 (1983; Zbl 0517.62053)], and, finally, the case of the density ratio of a maximal invariant is considered.
The book is completed by a reasonable number of references, an extensive list of symbols and an exhaustive subject index.
Reviewer: R.Schwabe (Berlin)

62A01 Foundations and philosophical topics in statistics
62-02 Research exposition (monographs, survey articles) pertaining to statistics
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
53C99 Global differential geometry
Full Text: Euclid