Annals of Statistics

Subjective Hierarchical Bayes Estimation of a Multivariate Normal Mean: On the Frequentist Interface

James O. Berger and Christian Robert

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In shrinkage estimation of a multivariate normal mean, the two dominant approaches to construction of estimators have been the hierarchical or empirical Bayes approach and the minimax approach. The first has been most extensively used in practice, because of its greater flexibility in adapting to varying situations, while the second has seen the most extensive theoretical development. In this paper we consider several topics on the interface of these approaches, concentrating, in particular, on the interface between hierarchical Bayes and frequentist shrinkage estimation. The hierarchical Bayes setup considered is quite general, allowing (and encouraging) utilization of subjective second stage prior distributions to represent knowledge about the actual location of the normal means. (The first stage of the prior is used, as usual, to model suspected relationships among the means.) We begin by providing convenient representations for the hierarchical Bayes estimators to be considered, as well as formulas for their associated posterior covariance matrices and unbiased estimators of matrical mean square error; these are typically proposed by Bayesians and frequentists, respectively, as possible "error matrices" for use in evaluating the accuracy of the estimators. These two measures of accuracy are extensively compared in a special case, to highlight some general features of their differences. Risks and various estimated risks or losses (with respect to quadratic loss) of the hierarchical Bayes estimators are also considered. Some rather surprising minimax results are established (such as one in which minimaxity holds for any subjective second stage prior of the mean), and the various risks and estimated risks are extensively compared. Finally, a conceptually trivial (but often calculationally difficult) method of verifying minimaxity is illustrated, based on numerical maximization of the unbiased estimator of risk (using certain convenient calculational formulas for hierarchical Bayes estimators), and is applied to an illustrative example.

Article information

Ann. Statist., Volume 18, Number 2 (1990), 617-651.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F15: Bayesian inference
Secondary: 62C20: Minimax procedures 62J07: Ridge regression; shrinkage estimators

Hierarchical Bayes estimation subjective second stage prior distributions posterior covariance matrix unbiased estimator of matricial mean square error frequentist risk posterior expected loss unbiased estimator of risk superharmonic distributions


Berger, James O.; Robert, Christian. Subjective Hierarchical Bayes Estimation of a Multivariate Normal Mean: On the Frequentist Interface. Ann. Statist. 18 (1990), no. 2, 617--651. doi:10.1214/aos/1176347619.

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