Shrinkage and Orthogonal Decomposition

ABSTRACT. The problem of estimating the mean of a multivariate normal distribution when the parameter space allows an orthogonal decomposition is discussed. Risk functions and lower bounds for a class of shrinkage estimators that includes Stein's estimator are derived, and an improvement on Stein's estimator that takes advantage of the orthogonal decomposition is introduced. Uniform asymptotics related to Pinsker's minimax risk is derived and we give conditions for attaining the lower risk bound. Special cases including regression and analysis of variance are discussed.