Computation of multivariate normal and \(t\) probabilities. (English) Zbl 1204.62088

Lecture Notes in Statistics 195. Berlin: Springer (ISBN 978-3-642-01688-2/pbk; 978-3-642-01689-9/ebook). viii, 124 p. (2009).
This monograph on the computation of multivariate normal and multivariate \(t\) probabilities consists of 6 chapters, Appendix A and Appendix B. Chapter 1 introduces the problem. After a brief description of the historical development some examples of 3-,5- and 8-dimensional cases are illustrated. Chapter 2 concentrates on bivariate and trivariate cases. Special regions of integration such as orthants, ellipsoids and hyperboloids are considered. Special correlation structures, such as diagonal, reduced rank and banded correlation matrices, are examined. Chapter 3 deals with approximation problems. Boole formula approximations, correlation matrix approximations, asymptotic expansions and other approximations are considered. Chapter 4 deals with approximating the integrals by (1) reparameterizations of spherical-radial transformations, the separation of variables method, variable reordering method and tridiagonal decomposition methods; (2) different integration methods such as Monte-Carlo and quasi Monte-Carlo methods, polynomial integration method, sub-region adaptive methods, sparse-grid methods, and other numerical integration methods.
Chapter 5 goes into miscellaneous topics, such as linear inequality constraints, singular distributions, integrals over k-dimensional cubes, and software such as R and MATLAB. Chapter 6 examines some applications such as (1) multiple comparison procedures in multiple testing of linear models and numerical computations of critical values; (2) Bayesian statistics and financial mathematics involving standardizing transformations to multivariate normal and t transformations, split transformations, etc. Appendix A describes R functions and Appendix B MATLAB functions.
On each topic the monograph goes into the complete development of the problem and the key results are also presented. Whenever needed, a few steps on software implementation are given. Software R and MATLAB are explained in detail. A list of websites for various computer programs and packages is also given. This will help the reader for further exploration of the topics.
This monograph is a valuable source of information for people involved in the computations of multivariate normal and multivariate t probabilities. The material is assembled and presented in a consistent easy-flowing format. In general, this is an excellent monograph and a valuable addition to the personal library of people working in the areas of multivariate statistical analysis.


62H10 Multivariate distribution of statistics
65C60 Computational problems in statistics (MSC2010)
62-04 Software, source code, etc. for problems pertaining to statistics
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
62H20 Measures of association (correlation, canonical correlation, etc.)
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