On estimating a common multivariate normal mean vector (Q1082012)

Let \(X_ 1,...,X_ n\) be a random sample of population \(N_ p(\mu,\Sigma_ X)\) and \(Y_ 1,...,Y_ n\) be a random sample of population \(N_ p(\mu,\Sigma_ Y)\). Assume the X-sample and Y-sample are independent. Consider the problem of estimating the common mean vector \(\mu\). When \(\Sigma_ X\) and \(\Sigma_ Y\) are known the unbiased estimator \[ T_ 3=\Sigma_ Y(\Sigma_ X+\Sigma_ Y)^{-1}\bar X+\Sigma_ X(\Sigma_ X+\Sigma_ Y)^{-1}\bar Y \] is better than \(\bar X\) and \(\bar Y\) in the sense that \(\Sigma_{\bar X}-\Sigma_{T_ 3}>0\) and \(\Sigma_{\bar Y}-\Sigma_{T_ 3}>0\). When \(\Sigma_ X\) and \(\Sigma_ Y\) are unknown it is logical to study the unbiased estimator \[ T_ 4=S_ Y(S_ X+S_ Y)^{-1}\bar X+S_ X(S_ X+S_ Y)^{- 1}\bar Y \] where \(S_ X\) and \(S_ Y\) are the sample covariance matrices of X and Y, respectively. In the paper the authors point out that neither \((\Sigma_{\bar X}-\Sigma_{T_ 4})\) nor \((\Sigma_{\bar Y}-\Sigma_{T_ 4})\) is positive semi-definite for all \((\Sigma_ X,\Sigma_ Y)\) for any n. They also give more general results of this negative type.