Estimating the Common Mean of Two Multivariate Normal Distributions

Abstract

Let $X_1, X_2$ be two $p \times 1$ multivariate normal random vectors and $S_1, S_2$ be two $p \times p$ Wishart matrices, where $X_1 \sim N_p(\xi, \sum_1), X_2 \sim N_p(\xi, \sum_2), S_1 \sim W_p(\sum_1, n)$ and $S_2 \sim W_p(\sum_2, n)$. We further assume that $X_1, X_2, S_1, S_2$ are stochastically independent. We wish to estimate the common mean $\xi$ with respect to the loss function $L = (\hat{\xi} - \xi)'(\sum^{-1}_1 + \sum^{-1}_2)(\hat{\xi} - \xi)$. By extending the methods of Stein and Haff, an alternative unbiased estimator to the usual generalized least squares estimator is obtained. However, the risk of this estimator is not available in closed form. A Monte Carlo swindle is used instead to evaluate its risk performance. The results indicate that the alternative estimator performs very favorably against the usual estimator.