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April 1998 Quadratic and inverse regressions for Wishart distributions
Gérard Letac, Hélène Massam
Ann. Statist. 26(2): 573-595 (April 1998). DOI: 10.1214/aos/1028144849


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.


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Gérard Letac. Hélène Massam. "Quadratic and inverse regressions for Wishart distributions." Ann. Statist. 26 (2) 573 - 595, April 1998.


Published: April 1998
First available in Project Euclid: 31 July 2002

zbMATH: 1073.62536
MathSciNet: MR1626071
Digital Object Identifier: 10.1214/aos/1028144849

Primary: 62H05
Secondary: 60E10

Rights: Copyright © 1998 Institute of Mathematical Statistics


Vol.26 • No. 2 • April 1998
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