Estimating the common mean of two multivariate normal distributions. (English) Zbl 0742.62055

Let \(X_ 1\sim N_ p(\xi,\Sigma_ 1)\), \(X_ 2\sim N_ p(\xi,\Sigma_ 2)\), \(S_ 1\sim W_ p(\Sigma_ 1,n)\) and \(S_ 2\sim W_ p(\Sigma_ 2,n)\) be independent. It is desired to estimate the common mean \(\xi\) with respect to the quadratic loss function \(L=(\hat\xi-\xi)'(\Sigma_ 1^{-1}+\Sigma_ 2^{-1})(\hat\xi-\xi)\), which is a natural symmetric extension of the one-sample case. The estimators are special cases of (1): \(B^{-1}\Phi BX_ 1+B^{-1}(I- \Phi)BX_ 2\), where \(B(S_ 1+S_ 2)B'=I\), \(BS_ 2B'=F=\hbox {diag}(f_ 1,\dots,f_ p)\) with \(f_ 1\geq\dots\geq f_ p\), and \(\Phi\) is a diagonal matrix depending on \(| B(X_ 1-X_ 2)|^ 2\) and \(F\).
If \(\Sigma_ 1\) and \(\Sigma_ 2\) are known, the best linear unbiased estimator of \(\xi\) with respect to \(L\) is \(\hat\xi^{BE}=(\Sigma_ 1^{-1}+\Sigma_ 2^{-1})^{-1}(\Sigma_ 1^{-1}X_ 1+\Sigma_ 2^{-1} X_ 2)\), and has risk \(p\). The usual estimator with \(\Sigma_ 1\) and \(\Sigma_ 2\) unknown is \(\hat\xi^{LS}\) with \(\Sigma_ i\) replaced by \(S_ i\), \(i=1,2\); this is the generalized least squares estimator that minimizes \(\Sigma_ i(X_ i-\hat\xi)'S_ i^{-1}(X_ i-\hat\xi)\). It is to be compared with the Stein-Haff estimator, obtained from (1) when \(\Phi=\Phi^{SH}\), a certain matrix defined in the paper.
“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”.


62H12 Estimation in multivariate analysis
62F10 Point estimation
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