ON THE CONSTRUCTION OF BAYES MINIMAX ESTIMATORS 1

Bayes estimation of the mean of a multivariate normal distribution is considered under quadratic loss. We show that, when particular spherical priors are used, the superharmonicity of the square root of the marginal density provides a viable method for constructing (possibly proper) Bayes (and admissible) minimax estimators. Examples illustrate the theory; most notably it is shown that a multivariate Student- t prior yields a proper Bayes minimax estimate.

[1]  Abdul L. Bello,et al.  Tables of Integrals, Series, and Products , 1995 .

[2]  B. Sansó,et al.  Near ignorance classes of log-concave priors for the location model , 1992 .

[3]  James O. Berger,et al.  Robust hierarchical Bayes estimation of exchangeable means , 1991 .

[4]  James O. Berger,et al.  Subjective Hierarchical Bayes Estimation of a Multivariate Normal Mean: On the Frequentist Interface , 1990 .

[5]  William D. Sudderth,et al.  Coherent Inference from Improper Priors and from Finitely Additive Priors , 1989 .

[6]  J. Berger.  Statistical Decision Theory and Bayesian Analysis , 1988 .

[7]  Tze Fen Li,et al.  A family of admissible minimax estimators of the mean of a multivariate, normal distribution , 1986 .

[8]  E. George.  Combining Minimax Shrinkage Estimators , 1986 .

[9]  J. Doob.  Classical potential theory and its probabilistic counterpart , 1984 .

[10]  C. Stein.  Estimation of the Mean of a Multivariate Normal Distribution , 1981 .

[11]  A. Dawid.  Posterior expectations for large observations , 1973 .

[12]  On the existence of proper Bayes minimax estimators of the mean of a multivariate normal distribution , 1972 .

[13]  L. Brown.  Admissible Estimators, Recurrent Diffusions, and Insoluble Boundary Value Problems , 1971 .

[14]  W. Strawderman.  Proper Bayes Minimax Estimators of the Multivariate Normal Mean , 1971 .

[15]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[16]  E. C. Titchmarsh,et al.  The Laplace Transform , 1991, Heat Transfer 1.