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August 2001 Improving on the MLE of a bounded normal mean
Éric Marchand, François Perron
Ann. Statist. 29(4): 1078-1093 (August 2001). DOI: 10.1214/aos/1013699994


We consider the problem of estimating the mean of a $p$-variate normal distribution with identity covariance matrix when the mean lies in a ball of radius $m$. It follows from general theory that dominating estimators of the maximum likelihood estimator always exist when the loss is squared error. We provide and describe explicit classes of improvements for all problems $(m, p)$. We show that,for small enough $m$, a wide class of estimators, including all Bayes estimators with respect to orthogonally invariant priors, dominate the maximum likelihood estimator. When $m$ is not so small, we establish general sufficient conditions for dominance over the maximum likelihood estimator. These include, when $m \le \sqrt{p}$, the Bayes estimator with respect to a uniform prior on the boundary of the parameter space. We also study the resulting Bayes estimators for orthogonally invariant priors and obtain conditions of dominance involving the choice of the prior. Finally, these Bayesian dominance results are further discussed and illustrated with examples, which include (1) the Bayes estimator for a uniform prior on the whole parameter space and (2) a new Bayes estimator derived from an exponential family of priors.


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Éric Marchand. François Perron. "Improving on the MLE of a bounded normal mean." Ann. Statist. 29 (4) 1078 - 1093, August 2001.


Published: August 2001
First available in Project Euclid: 14 February 2002

zbMATH: 1041.62016
MathSciNet: MR1869241
Digital Object Identifier: 10.1214/aos/1013699994

Primary: 62F10 , 62F15 , 62F30

Keywords: Bayes estimators , Langevin distribution , maximum likelihood estimator , modified Bessel function , monotone likelihood ratio , multivariate normal distribution , noncentral chi-square distribution , restricted parameter space , squared error loss

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 4 • August 2001
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