Matrix Variate DistributionsUseful in physics, economics, psychology, and other fields, random matrices play an important role in the study of multivariate statistical methods. Until now, however, most of the material on random matrices could only be found scattered in various statistical journals. Matrix Variate Distributions gathers and systematically presents most of the recent developments in continuous matrix variate distribution theory and includes new results. After a review of the essential background material, the authors investigate the range of matrix variate distributions, including: With its inclusion of new results, Matrix Variate Distributions promises to stimulate further research and help advance the field of multivariate statistical analysis. |
Contents
PRELIMINARIES | 1 |
MATRIX VARIATE NORMAL DISTRIBUTION | 55 |
WISHART DISTRIBUTION | 87 |
MATRIX VARIATE tDISTRIBUTION | 133 |
MATRIX VARIATE BETA DISTRIBUTIONS | 165 |
MATRIX VARIATE DIRICHLET DISTRIBUTIONS | 199 |
DISTRIBUTION OF QUADRATIC FORMS | 225 |
MISCELLANEOUS DISTRIBUTIONS | 279 |
GENERAL FAMILIES OF MATRIX VARIATE | 311 |
GLOSSARY OF NOTATIONS AND ABBREVIATIONS | 331 |
| 343 | |
| 364 | |
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Common terms and phrases
ar+1 b₁ br+1 characteristic function constant matrix COROLLARY defined denoted density function derived det(A det(Ip gamma Gupta H₁ Hence independently distributed integral Jacobian joint density joint p.d.f. Khatri Laplace transform Lemma lower triangular matrix matrix of rank matrix variate beta matrix variate Dirichlet matrix variate normal matrix variate t-distribution matrix with positive Ms(Z multivariate n₁ noncentral matrix variate noncentral Wishart nonsingular obtained Olkin orthogonal matrix p.d.f. is given partition positive diagonal elements probability element Proof Prove Theorem quadratic forms random matrix rank(A Re(a Re(b Re(h result follows right spherical S₁ Stiefel manifold stochastically independent Substituting symmetric matrix symmetric positive definite THEOREM tr(A type II distribution U₁ V₁ variate beta type vec(X vector Wishart distribution Wp(n X₁ Y₁ zonal polynomials ΑΣΑ Γρ(α Γρ(η Σ & Ψ ΣΣ ΣΨ Гр(а
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Page 351 - Reprinted in Statistical Inference in Elliptically Contoured and Related Distributions (KT Fang and TW Anderson, eds.), Allerton Press, New York.