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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”.

MSC:

62H12 Estimation in multivariate analysis
62F10 Point estimation
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