## 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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