Shrinkage estimation with a matrix loss function

Consider estimating an n×p matrix of means Θ, say, from an n×p matrix of observations X, where the elements of X are assumed to be independently normally distributed with E(xij)=θij and constant variance, and where the performance of an estimator is judged using a p×p matrix quadratic error loss function. A matrix version of the James-Stein estimator is proposed, depending on a tuning constant a. It is shown to dominate the usual maximum likelihood estimator for some choices of a when n≥3. This result also extends to other shrinkage estimators and settings.