Open Access
September, 1980 Eigenfunctions of Expected Value Operators in the Wishart Distribution
H. B. Kushner, Morris Meisner
Ann. Statist. 8(5): 977-988 (September, 1980). DOI: 10.1214/aos/1176345136

Abstract

Let $(X_{1,m}, X_{2,m}, \cdots X_{k,m}), 1 \leqslant m \leqslant n,$ be a sample of size $n$ from the $k$ dimensional normal distribution with mean vector $\mu$ and covariance matrix $\Sigma.$ Let $V = (\nu_{ij}), 1 \leqslant i, j \leqslant k,$ denote the symmetric scatter matrix where $v_{ij} = \sum_m(X_{i,m} - \mu_i)(X_{j,m} - \mu_j).$ The problem posed is to characterize the eigenfunctions of the expectation operators of the Wishart distribution, i.e., those scalar valued functions, $f(V),$ such that $E(f(V)) = \lambda_{n,k}f(\Sigma).$ If $f$ is an eigenfunction then (a) for nonsingular $T,f(T'VT)$ is an eigenfunction and (b) for integral $p, |V|^{p/2}f(V)$ is an eigenfunction. For $k \leqslant 2,$ a complete solution of the problem is given. For $k = 1$ the functions $f(v) = cv^\alpha$ are the only eigenfunctions. For $k = 2,$ a function $f$ is an eigenfunction if and only if (i) $f$ is homogeneous and (ii) $4 \frac{\partial^2 f}{\partial v_{22}\partial v_{11}} - \frac{\partial^2 f}{\partial^2v_{21}} = C|V|^{-1}f.$ A representation of eigenfunctions is given in terms of sums of associated Legendre functions. Relationships between eigenfunctions and harmonic functions are indicated. Any homogeneous polynomial is proved to be a linear combination of polynomial eigenfunctions.

Citation

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H. B. Kushner. Morris Meisner. "Eigenfunctions of Expected Value Operators in the Wishart Distribution." Ann. Statist. 8 (5) 977 - 988, September, 1980. https://doi.org/10.1214/aos/1176345136

Information

Published: September, 1980
First available in Project Euclid: 12 April 2007

zbMATH: 0446.62046
MathSciNet: MR585697
Digital Object Identifier: 10.1214/aos/1176345136

Subjects:
Primary: 62H99
Secondary: 45C05‎

Keywords: Eigenfunctions , expectation operators , homogeneous functions , second order linear partial differential equation , Wishart distribution

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 5 • September, 1980
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