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The variational form of certain Bayes estimators. (English) Zbl 0739.62046
A general representation is obtained for the formal Bayes estimator of a parameter matrix. The author assumes 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 this 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, the author gives the VFBE of several Pearson means.
A certain emphasis is placed on the problem of estimating the covariance matrix. For that problem, his 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 Stein’s estimator. 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.

MSC:
62H12 Estimation in multivariate analysis
62F15 Bayesian inference
62C99 Statistical decision theory
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