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March, 1964 Multivariate Beta Distributions and Independence Properties of the Wishart Distribution
Ingram Olkin, Herman Rubin
Ann. Math. Statist. 35(1): 261-269 (March, 1964). DOI: 10.1214/aoms/1177703748


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.


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Ingram Olkin. Herman Rubin. "Multivariate Beta Distributions and Independence Properties of the Wishart Distribution." Ann. Math. Statist. 35 (1) 261 - 269, March, 1964.


Published: March, 1964
First available in Project Euclid: 27 April 2007

zbMATH: 0128.14002
MathSciNet: MR160297
Digital Object Identifier: 10.1214/aoms/1177703748

Rights: Copyright © 1964 Institute of Mathematical Statistics

Vol.35 • No. 1 • March, 1964
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