zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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
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)