Improved multivariate normal mean estimation with unknown covariance when p is greater than n

Abstract

We consider the problem of estimating the mean vector of a p-variate normal $(\theta ,\Sigma )$ distribution under invariant quadratic loss, $(\delta -\theta {)}^{\prime }{\Sigma }^{-1}(\delta -\theta )$, 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.

Published in at http://dx.doi.org/10.1214/12-AOS1067 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Regularized estimation of large covariance matrices
• Sparse permutation invariant covariance estimation
• Sparse estimation of large covariance matrices via a nested Lasso penalty
• Shrinkage Estimation in Multilevel Normal Models