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January, 1973 Generalized Bayes Minimax Estimators of the Multivariate Normal Mean with Unknown Covariance Matrix
Pi-Erh Lin, Hui-Liang Tsai
Ann. Statist. 1(1): 142-145 (January, 1973). DOI: 10.1214/aos/1193342390

Abstract

Let $\mathbf{X}$ be a $p$-variate $(p \geqq 3)$ vector normally distributed with mean $\mathbf{\theta}$ and covariance matrix $\Sigma$, positive definite but unknown. Let $A$ be a $p \times p$ Wishart matrix with parameters $(n, \Sigma)$, independent of $\mathbf{X}$. To estimate $\mathbf{\theta}$ relative to quadratic loss function $(\hat{\mathbf{\theta}} - \mathbf{\theta})'\Sigma^{-1}(\hat{\mathbf{\theta}} - \mathbf{\theta})$, we obtain a family of minimax estimators $\mathbf{\delta}(\mathbf{X}, \mathbf{A})$ based on $\mathbf{X}$ and $\mathbf{A}$ through $\mathbf{X}$ and $\mathbf{X}'\mathbf{A}^{-1}\mathbf{X}$. It is shown that there are minimax estimators of the form $\mathbf{\delta}(\mathbf{X}, \mathbf{A})$ which are also generalized Bayes. A special case where $\Sigma = \sigma^2\mathbf{I}$ is also considered.

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Pi-Erh Lin. Hui-Liang Tsai. "Generalized Bayes Minimax Estimators of the Multivariate Normal Mean with Unknown Covariance Matrix." Ann. Statist. 1 (1) 142 - 145, January, 1973. https://doi.org/10.1214/aos/1193342390

Information

Published: January, 1973
First available in Project Euclid: 25 October 2007

zbMATH: 0254.62006
MathSciNet: MR331573
Digital Object Identifier: 10.1214/aos/1193342390

Rights: Copyright © 1973 Institute of Mathematical Statistics

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Vol.1 • No. 1 • January, 1973
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