A Characterization of the Wishart Distribution

Abstract

It is known that if $X$ and $Y$ are independent random variables having a Gamma distribution with parameters $(\theta, n)$ and $(\theta, m)$, i.e., with density function $p(x, \theta, n) = \frac{\theta^{n/2}x^{n/2 - 1}e^{-(\frac{1}{2})\theta x}}{2^{n/2}\Gamma(n/2)},\quad 0 < x, \theta; 1 \leqq n,$ then $X + Y$ and $X/(X + Y)$, or equivalently $X/Y$, are statistically independent. Lukacs [1] proved that this independence property characterizes the Gamma distribution, namely, if $X$ and $Y$ are two nondegenerate positive random variables, and if $X + Y$ is independent of $X/(X + Y)$, or equivalently of $X/U$, then $X$ and $Y$ have a Gamma distribution with the same scale parameter. In the present paper we present an extension to the case where $U$ and $V$ are symmetric positive definite matrices having a Wishart distribution. A number of difficulties are encountered in the generalization. First, there is no natural extension of a ratio, and we consider $Z = W^{-1}UW'$ $^{-1}$, where the "square root" $W = (U + V)^{\frac{1}{2}}$ is any factorization $WW' = (U + V)$. In the matrix case $Z$ is not a function of $V^{-\frac{1}{2}}UV^{-\frac{1}{2}}$ as was true in one dimension, and indeed if $U$ and $V$ are independent random matrices having a Wishart distribution, $U + V$ and $V^{-\frac{1}{2}}UV^{-\frac{1}{2}}$ need not be statistically independent, depending on which square root is chosen. This aspect will be treated in another paper. In the univariate case it is relatively straightforward to generate differential equations by differentiating under the expectation sign, but this is no longer true since the elements of $(U + V)^{\frac{1}{2}}$ do not bear a simple relation to the elements of $(U + V)$, and it is this point which leads to the difficulties in the proof. The characterization theorem is stated in Section 2. In Section 3 the differential equation is set up, and is solved in Section 4. The authors are indebted to Martin Fox for his comments and suggestions.