Estimation of a Mean Vector in a Two-Sample Problem

We consider the problem of estimating a p-dimensional vector μ1 based on independent variables X1, X2, and U, where X1 is Np(μ1, σ2Σ1), X2 is Np(μ2, σ2Σ2), and U is σ2χ2n (Σ1 and Σ2 are known). A family of minimax estimators is proposed. Some of these estimators can be obtained via Bayesian arguments as well. Comparisons between our results and the one of Ghosh and Sinha (1988, J. Multivariate Anal.27 206-207) are presented.