Quadratic and inverse regressions for Wishart distributions
成果类型：
Article
署名作者：
Letac, G; Massam, H
署名单位：
Universite de Toulouse; Universite Toulouse III - Paul Sabatier; York University - Canada
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
发表日期：
1998
页码：
573-595
关键词：
摘要：
If U and V are independent random variables which are gamma distributed with the same scale parameter, then there exist a and b in R such that E(U | U + V) = a(U + V) and 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 E(U-1) exists, there exists c in R such that E(U-1 | U + V) = c(U + V)(-1) Wesolowski 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.