The Annals of Statistics

The Variational Form of Certain Bayes Estimators

L. R. Haff

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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.

Article information

Ann. Statist., Volume 19, Number 3 (1991), 1163-1190.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62H12: Estimation
Secondary: 62C99: None of the above, but in this section

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


Haff, L. R. The Variational Form of Certain Bayes Estimators. Ann. Statist. 19 (1991), no. 3, 1163--1190. doi:10.1214/aos/1176348244.

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