The Annals of Statistics

Multivariate Empirical Bayes and Estimation of Covariance Matrices

Bradley Efron and Carl Morris

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The problem of estimating several normal mean vectors in an empirical Bayes situation is considered. In this case, it reduces to the problem of estimating the inverse of a covariance matrix in the standard multivariate normal situation using a particular loss function. Estimators which dominate any constant multiple of the inverse sample covariance matrix are presented. These estimators work by shrinking the sample eigenvalues toward a central value, in much the same way as the James-Stein estimator for a mean vector shrinks the maximum likelihood estimators toward a common value. These covariance estimators then lead to a class of multivariate estimators of the mean, each of which dominates the maximum likelihood estimator.

Article information

Ann. Statist., Volume 4, Number 1 (1976), 22-32.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F10: Point estimation
Secondary: 62C99: None of the above, but in this section

Multivariate empirical Bayes Stein's estimator minimax estimation mean of a multivariate normal distribution estimating a covariance matrix James-Stein estimator simultaneous estimation combining estimates


Efron, Bradley; Morris, Carl. Multivariate Empirical Bayes and Estimation of Covariance Matrices. Ann. Statist. 4 (1976), no. 1, 22--32. doi:10.1214/aos/1176343345.

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