Shrinkage and orthogonal decomposition

The problem of estimating the mean of a multivariate normal distribution wh en the parameter space allows an orthogonal decomposition is discussed. Ris k functions and lower bounds for a class of shrinkage estimators that inclu des 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 condi tions for attaining the lower risk bound. Special cases including regressio n and analysis of variance are discussed.