Minimax Multiple Shrinkage Estimation

For the canonical problem of estimating a multivariate normal mean under squared-error-loss, this article addresses the problem of selecting a minimax shrinkage estimator when vague or conflicting prior information suggests that more than one estimator from a broad class might be effective. For this situation a new class of alternative estimators, called multiple shrinkage estimators, is proposed. These estimators use the data to emulate the behavior and risk properties of the most effective estimator under consideration. Unbiased estimates of risk and sufficient conditions for minimaxity are provided. Bayesian motivations link this construction to posterior means of mixture priors. To illustrate the theory, minimax multiple shrinkage Stein estimators are constructed which can adaptively shrink the data towards any number of points or subspaces.