Non-Optimality of Preliminary-Test Estimators for the Mean of a Multivariate Normal Distribution

Estimation-preceded-by-testing is studied in the context of estimating the mean vector of a multivariate normal distribution with quadratic loss. It is shown that although there are parameter values for which the risk of a preliminary-test estimator is less than that of the usual estimator, there are also values for which its risk exceeds that of the usual estimator, and that it is dominated by the positive-part version of the Stein-James estimator. The results apply to preliminary-test estimators corresponding to any linear hypothesis concerning the mean vector, e.g., an hypothesis in a regression model. The case in which the covariance matrix of the multi-normal distribution is known up to a multiplicative constant and the case in which it is completely unknown are treated.