## The Annals of Mathematical Statistics

### Multivariate Beta Distributions and Independence Properties of the Wishart Distribution

#### 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.

#### Article information

Source
Ann. Math. Statist., Volume 35, Number 1 (1964), 261-269.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177703748

Digital Object Identifier
doi:10.1214/aoms/1177703748

Mathematical Reviews number (MathSciNet)
MR160297

Zentralblatt MATH identifier
0128.14002

JSTOR