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December 2012 Improved multivariate normal mean estimation with unknown covariance when $p$ is greater than $n$
Didier Chételat, Martin T. Wells
Ann. Statist. 40(6): 3137-3160 (December 2012). DOI: 10.1214/12-AOS1067

Abstract

We consider the problem of estimating the mean vector of a $p$-variate normal $(\theta,\Sigma)$ distribution under invariant quadratic loss, $(\delta-\theta)'\Sigma^{-1}(\delta-\theta)$, when the covariance is unknown. We propose a new class of estimators that dominate the usual estimator $\delta^{0}(X)=X$. The proposed estimators of $\theta$ depend upon $X$ and an independent Wishart matrix $S$ with $n$ degrees of freedom, however, $S$ is singular almost surely when $p>n$. The proof of domination involves the development of some new unbiased estimators of risk for the $p>n$ setting. We also find some relationships between the amount of domination and the magnitudes of $n$ and $p$.

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Didier Chételat. Martin T. Wells. "Improved multivariate normal mean estimation with unknown covariance when $p$ is greater than $n$." Ann. Statist. 40 (6) 3137 - 3160, December 2012. https://doi.org/10.1214/12-AOS1067

Information

Published: December 2012
First available in Project Euclid: 22 February 2013

zbMATH: 1296.62048
MathSciNet: MR3097972
Digital Object Identifier: 10.1214/12-AOS1067

Subjects:
Primary: 62F10
Secondary: 62C20 , 62H12

Keywords: Covariance estimation , invariant quadratic loss , James–Stein estimation , large-$p$–small-$n$ problems , location parameter , minimax estimation , Moore–Penrose inverse , risk function , singular Wishart distribution

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 6 • December 2012
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