IMPROVED MULTIVARIATE NORMAL MEAN ESTIMATION WITH UNKNOWN COVARIANCE WHEN p IS GREATER THAN n

We consider the problem of estimating the mean vector of a p-variate normal (0, .) distribution under invariant quadratic loss, (.-0)'..¹(.-0), when the covariance is unknown. We propose a new class of estimators that dominate the usual estimator .. (X) = X. The proposed estimators of . 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.