Open Access
September, 1991 The Variational Form of Certain Bayes Estimators
L. R. Haff
Ann. Statist. 19(3): 1163-1190 (September, 1991). DOI: 10.1214/aos/1176348244


A general representation is obtained for the formal Bayes estimator of a parameter matrix. We assume that the prior distribution is symmetric in some sense, but it is not specified otherwise. The formal Bayes risk is minimized subject to order constraints by a variational technique; hence our representation is called "the variational form of the Bayes estimator" (VFBE). The VFBE is used to obtain estimators that have good frequency properties relative to the usual estimators. Such estimators are obtained for the mean vector and covariance matrix of a multivariate normal distribution. Also, for possibly nonnormal data, we give the VFBE of several Pearson means. A certain emphasis is placed on the problem of estimating the covariance matrix. For that problem, our constrained optimization provides an estimator with very good properties: Its eigenvalues are in the proper order, and they are not as distorted as those in the sample covariance matrix. The VFBE for the covariance matrix is related to an estimator of Stein. Of the two, the VFBE deals with order relations in a more natural way; that is, it is more criterion dependent. In addition, it is easier to compute than Stein's estimator, and a brief Monte Carlo simulation indicates that it has better risk properties as well.


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L. R. Haff. "The Variational Form of Certain Bayes Estimators." Ann. Statist. 19 (3) 1163 - 1190, September, 1991.


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

zbMATH: 0739.62046
MathSciNet: MR1126320
Digital Object Identifier: 10.1214/aos/1176348244

Primary: 62H12
Secondary: 62C99

Keywords: Covariance matrix , eigenvalue distortion in sample covariance matrix , estimation of eigenvalues , Euler equations , Mean vector , orthogonally invariant estimators , Pearson curves , spherically symmetric estimators , unbiased estimation of risk , variational form of Bayes estimators

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 3 • September, 1991
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