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March, 1991 Estimating the Common Mean of Two Multivariate Normal Distributions
Wei-Liem Loh
Ann. Statist. 19(1): 297-313 (March, 1991). DOI: 10.1214/aos/1176347983


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.


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Wei-Liem Loh. "Estimating the Common Mean of Two Multivariate Normal Distributions." Ann. Statist. 19 (1) 297 - 313, March, 1991.


Published: March, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0742.62055
MathSciNet: MR1091852
Digital Object Identifier: 10.1214/aos/1176347983

Primary: 62F10
Secondary: 62C99

Keywords: Common mean , equivariant estimation , unbiased estimate of risk , Wishart distribution

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 1 • March, 1991
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