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July, 1977 Minimax Estimation of a Normal Mean Vector for Arbitrary Quadratic Loss and Unknown Covariance Matrix
J. Berger, M. E. Bock, L. D. Brown, G. Casella, L. Gleser
Ann. Statist. 5(4): 763-771 (July, 1977). DOI: 10.1214/aos/1176343898

Abstract

Let $X$ be an observation from a $p$-variate normal distribution $(p \geqq 3)$ with mean vector $\theta$ and unknown positive definite covariance matrix $\not\Sigma$. It is desired to estimate $\theta$ under the quadratic loss $L(\delta, \theta, \not\Sigma) = (\delta - \theta)^tQ(\delta - \theta)/\operatorname{tr} (Q\not\Sigma)$, where $Q$ is a known positive definite matrix. Estimators of the following form are considered: $\delta^c(X, W) = (I - c\alpha Q^{-1}W^{-1}/(X^tW^{-1}X))X,$ where $W$ is a $p \times p$ random matrix with a Wishart $(\not\Sigma, n)$ distribution (independent of $X$), $\alpha$ is the minimum characteristic root of $(QW)/(n - p - 1)$ and $c$ is a positive constant. For appropriate values of $c, \delta^c$ is shown to be minimax and better than the usual estimator $\delta^0(X) = X$.

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J. Berger. M. E. Bock. L. D. Brown. G. Casella. L. Gleser. "Minimax Estimation of a Normal Mean Vector for Arbitrary Quadratic Loss and Unknown Covariance Matrix." Ann. Statist. 5 (4) 763 - 771, July, 1977. https://doi.org/10.1214/aos/1176343898

Information

Published: July, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0356.62009
MathSciNet: MR443156
Digital Object Identifier: 10.1214/aos/1176343898

Subjects:
Primary: 62C99
Secondary: 62F10 , 62H99

Keywords: ‎mean‎ , minimax , normal , quadratic loss , risk function , unknown covariance matrix , Wishart

Rights: Copyright © 1977 Institute of Mathematical Statistics

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Vol.5 • No. 4 • July, 1977
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