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. |
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This work is the most complete reference in existence for matrix-variate distributions.
Contents
| 1 | |
| 55 | |
| 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 |
REFERENCES | 343 |
SUBJECT INDEX | 364 |
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Common terms and phrases
Chapter characteristic function completes COROLLARY defined denoted density function depend derived det(A det(I det(S det(X expression follows Further gamma give given Gupta Hence hypergeometric function independently distributed integral inverted Jacobian joint density Khatri kind Laplace transform Lemma Let X lower triangular marginal matrix variate matrix variate beta multivariate multivariate statistical n x n Non(M noncentral nonsingular normal distribution Note obtained orthogonal orthogonal matrix p x n p x p parameters partition polynomials Proof properties Prove Theorem quadratic forms random matrices rank rank(A Re(a reduces representation respectively result result follows roots similar spherical square statistical studied Substituting symmetric matrices symmetric matrix symmetric positive definite Tr(a transformation type II distribution upper values variate beta type vec(X vector Wishart distribution write
Popular passages
Page 179 - T'T, where T is an upper triangular matrix with positive diagonal elements, and let U = (T') ~ *QjT~ l.
Page 18 - ¡R!(X)dX is defined as the iterated integral of f(X) with respect to each element of X separately over a region R in the space defined by the simplex bounding the ranges of the elements of X.
Page 3 - A is positive definite (positive semidefinite) if and only if all the characteristic roots of A are positive (non-negative).
Page 39 - A^(RS) from (1.6.15) in (1.7.1), changing the order of integration and integrating with respect to R we get / etr(Z) det(Z)-^-5(p+1)cK(/p - SZ-l) dZ.
Page 19 - Fm(5) is the generalization to matrix variables of the Eulerian integral of the second kind. We have now proved (1.1) for real Z; it follows for complex Z by analytic continuation. Since Re Z > 0, det Z И 0 and (det Z)' is well defined by continuation.
Page 351 - Reprinted in Statistical Inference in Elliptically Contoured and Related Distributions (KT Fang and TW Anderson, eds.), Allerton Press, New York.
Page 42 - ... different outcomes which are not deterministically predictable. Instead the outcomes obey certain conditions of statistical regularity.
Page 32 - Distributional results of random matrices are often derived in terms of hypergeometric functions of matrix arguments.
Bibliographic information
| Title | Matrix Variate Distributions Volume 104 of Monographs and Surveys in Pure and Applied Mathematics |
| Authors | A K Gupta, D K Nagar |
| Edition | illustrated |
| Publisher | CRC Press, 1999 |
| ISBN | 1584880465, 9781584880462 |
| Length | 384 pages |
| Subjects | › › Mathematics / Applied Mathematics / Probability & Statistics / General |
| Export Citation | BiBTeX EndNote RefMan |