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January, 1976 Families of Minimax Estimators of the Mean of a Multivariate Normal Distribution
Bradley Efron, Carl Morris
Ann. Statist. 4(1): 11-21 (January, 1976). DOI: 10.1214/aos/1176343344


Ever since Stein's result, that the sample mean vector $\mathbf{X}$ of a $k \geqq 3$ dimensional normal distribution is an inadmissible estimator of its expectation $\mathbf{\theta}$, statisticians have searched for uniformly better (minimax) estimators. Unbiased estimators are derived here of the risk of arbitrary orthogonally-invariant and scale-invariant estimators of $\mathbf{\theta}$ when the dispersion matrix $\sum$ of $\mathbf{X}$ is unknown and must be estimated. Stein obtained this result earlier for known $\mathbf{\sum}$. Minimax conditions which are weaker than any yet published are derived by finding all estimators whose unbiased estimate of risk is bounded uniformly by $k$, the risk of $\mathbf{X}$. One sequence of risk functions and risk estimates applies simultaneously to the various assumptions about $\mathbf{\sum}$, resulting in a unified theory for these situations.


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Bradley Efron. Carl Morris. "Families of Minimax Estimators of the Mean of a Multivariate Normal Distribution." Ann. Statist. 4 (1) 11 - 21, January, 1976.


Published: January, 1976
First available in Project Euclid: 12 April 2007

zbMATH: 0322.62010
MathSciNet: MR403001
Digital Object Identifier: 10.1214/aos/1176343344

Primary: 62F10
Secondary: 62C99

Keywords: estimation , mean of a multivariate normal distribution , minimax estimators , risk of invariant estimators , Stein's estimator

Rights: Copyright © 1976 Institute of Mathematical Statistics


Vol.4 • No. 1 • January, 1976
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