Showing papers by "Ingram Olkin published in 1964"

Multivariate Beta Distributions and Independence Properties of the Wishart Distribution

Abstract: 1. Summary and introduction. If X and Y are independent random variables having chi-square distributions with n and m degrees of freedom, respectively, then except for constants, X/Y and X/(X + Y) are distributed as F and Beta variables. In the multivariate case, the Wishart distribution plays the role of the chi-square distribution. There is, however, no single natural generalization of a ratio in the multivariate case. In this paper several generalizations which lead to multivariate analogs of the Beta or F distribution are given. Some of these distributions arise naturally from a consideration of the sufficient statistic or maximal invariant in various multivariate problems, e.g., (i) testing that k normal populations are identical [1], p. 251, (ii) multivariate analysis of variance tests [9], (iii) multivariate slippage problems [4], p. 321. Although several of the results may be known as folklore, they have not been explicitly stated. Other of the distributions obtained are new. Intimately related to some of the distributional problems is the independence of certain statistics, and results in this direction are also given. 2. Notation a,nd comments. If V and W are symmetric matrices, V > W means that V - W is positive definite. I, denotes the identity of order p; the subscript is omitted when the dimensionality is clear from the context. We write etr A to mean exp tr A. X - Y means that X and Y have the same distribution. V 'W (, p, n) means that V is a p X p symmetric matrix whose p(p + 1)/2 elements are random variables having a Wishart distribution with (non-degenerate) covariance matrix =_ A-' and n degrees of freedom (n _ p assumed throughout), i.e., with density function.