The Annals of Statistics

Quadratic and inverse regressions for Wishart distributions

Gérard Letac and Hélène Massam

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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.

Article information

Source
Ann. Statist., Volume 26, Number 2 (1998), 573-595.

Dates
First available in Project Euclid: 31 July 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1028144849

Digital Object Identifier
doi:10.1214/aos/1028144849

Mathematical Reviews number (MathSciNet)
MR1626071

Zentralblatt MATH identifier
1073.62536

Subjects
Primary: 62H05: Characterization and structure theory
Secondary: 60E10: Characteristic functions; other transforms

Keywords
Natural exponential families Wishart distributions Jordan algebras conditional moments

Citation

Letac, Gérard; Massam, Hélène. Quadratic and inverse regressions for Wishart distributions. Ann. Statist. 26 (1998), no. 2, 573--595. doi:10.1214/aos/1028144849. https://projecteuclid.org/euclid.aos/1028144849


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