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, $(\delta-\theta)'\Sigma^{-1}(\delta-\theta)$, 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.