Minimax Estimation of a Normal Mean Vector for Arbitrary Quadratic Loss and Unknown Covariance Matrix

Let X be an observation from a p-variate normal distribution (p ≧ 3) with mean vector θ and unknown positive definite covariance matrix Σ̸. It is desired to estimate θ under the quadratic loss L(δ,θ,Σ̸)=(δ−θ)tQ(δ−θ)/tr(QΣ̸), where Q is a known positive definite matrix. Estimators of the following form are considered: δc(X,W)=(I−cαQ−1W−1/(XtW−1X))X, where W is a p × p random matrix with a Wishart (Σ̸,n) distribution (independent of X), α is the minimum characteristic root of (QW)/( n−p−1) and c is a positive constant. For appropriate values of c,δc is shown to be minimax and better than the usual estimator δ0(X)=X.