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 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
a₁ ar+1 b₁ br+1 characteristic function confluent hypergeometric function constant matrix COROLLARY defined denoted density function derived det(A det(I det(S Dirichlet distribution DTp,m(n F₁ 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 n₂ noncentral matrix variate noncentral Wishart nonsingular obtained Olkin orthogonal matrix p.d.f. is given P+¹ p₁ partition positive diagonal elements Proof Prove Theorem quadratic forms random matrix rank(A Re(a Re(b Re(h result follows S₁ S₂ Stiefel manifold stochastically independent Substituting symmetric matrix symmetric positive definite T₁ THEOREM type II distribution U₁ V₁ variate beta type vec(X vec(Y vector W₁ Wishart distribution Wp(n X₁ Y₁ zonal polynomials ΣΣ
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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.