Multivariate Beta Distributions and Independence Properties of the Wishart Distribution

TLDR

The Wishart distribution generalizes the chi‑square to multivariate settings, yet a natural multivariate analogue of a ratio such as X/Y is not obvious. This work introduces several generalizations that produce multivariate analogues of the Beta or F distribution. These distributions arise from examining sufficient statistics or maximal invariants in problems like testing equality of k normal populations, multivariate ANOVA, and slippage tests. The authors present both folklore results and new distributions, and establish independence properties of certain statistics.

Abstract

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.