A Family of Minimax Estimators of the Mean of a Multivariate Normal Distribution

Abstract

Ever since Stein's result, that the sample mean vector X of a k 2 3 dimensional normal distribution is an inadmissible estimator of its expectation @, statisticians have searched for uniformly better (minimax) estimators. Unbiased estimators are derived here of the risk of arbitrary orthogonally-invariant and scale-invariant estimators of @ when the dispersion matrix Z of X is unknown and must be estimated. Stein obtained this result earlier for known Z. Minimax conditions which are weaker than any yet published are derived by finding all estimators whose unbiased estimate of risk is bounded uniformly by k, the risk of X. One sequence of risk functions and risk estimates applies simultaneously to the various assumptions about 2, resulting in a unified theory for these situations.