Quadratic and inverse regressions for Wishart distributions

Abstract

If $U$ and $V$ are independent random variables which are gamma distributed with the same scale parameter, then there exist $a$ and $b$ in $\mathbb{R}$ such that $$\mathbb{E}(U|U + V) = a(U + V)$$ and $$\mathbb{E}(U^2|U + V) = b(U + V)^2$$. This, in fact, is characteristic of gamma distributions. Our paper extends this property to the Wishart distributions in a suitable way, by replacing the real number $U^2$ by a pair of quadratic functions of the symmetric matrix $U$. This leads to a new characterization of the Wishart distributions, and to a shorter proof of the 1962 characterization given by Olkin and Rubin. Similarly, if $\mathbb{E}(U^{-1})$ exists, there exists $c$ in $\mathbb{R}$ such that $$\mathbb{E}(U^{-1}|U + V) = c(U + V)^{-1}$$. Wesołowski has proved that this also is characteristic of gamma distributions. We extend it to the Wishart distributions. Finally, things are explained in the modern framework of symmetric cones and simple Euclidean Jordan algebras.