Simultaneous Estimation of Independent Normal Mean Vectors with Unknown Covariance Matrices

Based on independent samples from several multivariate normal populations, possibly of different dimensions, the problem of simultaneous estimation of the mean vectors is considered assuming that the covariance matrices are unknown. Two loss functions, the sum of usual quadratic losses and the sum of arbitrary quadratic losses, are used. A class of minimax estimators generalizing the James-Stein estimator is obtained. It is shown that these estimators improve the usual set of sample mean vectors uniformly under the sum of quadratic losses. This result is extended to the sum of arbitrary quadratic losses under some restrictions on the covariance matrices.