The Annals of Statistics

On the construction of Bayes minimax estimators

Dominique Fourdrinier, William E. Strawderman, and Martin T. Wells

Full-text: Open access


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.

Article information

Ann. Statist., Volume 26, Number 2 (1998), 660-671.

First available in Project Euclid: 31 July 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C20: Minimax procedures 62C15: Admissibility 62C10: Bayesian problems; characterization of Bayes procedures
Secondary: 62A15

Bayes estimate minimaxity multivariate normal mean proper Bayes quadratic loss superharmonic functions


Fourdrinier, Dominique; Strawderman, William E.; Wells, Martin T. On the construction of Bayes minimax estimators. Ann. Statist. 26 (1998), no. 2, 660--671. doi:10.1214/aos/1028144853.

Export citation


  • [1] Alam, K. (1973). A family of admissible minimax estimators of the mean of a multivariate normal distribution. Ann. Statist. 1 517-525.
  • [2] Angers, J. F. and Berger, J. O. (1991). Robust hierarchical Bay es estimation of exchangeable means. Canad. J. Statist. 19 39-56.
  • [3] Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analy sis. Springer, New York.
  • [4] Berger, J. O. and Robert, C. (1990). Subjective hierarchical Bay es estimation of a multivariate normal mean: on the frequentist interface. Ann. Statist. 18 617-651.
  • [5] Brown, L. D. (1971). Admissible estimators, recurrent diffusions and insoluble boundary value problems. Ann. Math. Statist. 42 855-903.
  • [6] Chance, B. L. and Wells, M. T. (1994). Characterizing hierarchical model behavior. In Proceedings of the 26th Sy mposium on the Interface. Computationally Intensive Statistical Methods (J. Sall and A. Lehman, eds.) 231-239.
  • [7] Dawid, A. P. (1973). Posterior expectations for large observations. Biometrika 60 664-667.
  • [8] Dellacherie, C. and Meyer, P.A. (1978). Probabilities and Potential. North-Holland, Amsterdam.
  • [9] Doob, J. L. (1984). Classical Potential Theory and Its Probabilistic Counterpart. Springer, New York.
  • [10] Faith, R. E. (1978). Minimax Bay es point estimators of a multivariate normal mean. J. Multivariate Analy sis 8 372-379.
  • [11] Fourdrinier, D. and Wells, M. T. (1996). Spherically sy mmetric Bay es estimators for a linear subspace of a normal law. In Bayesian Statistics 5 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 569-579. Oxford Univ. Press.
  • [12] George, E. I. (1986). Combining minimax shrinkage estimators. J. Amer. Statist. Assoc. 81 437-445.
  • [13] George, E. I. (1986). Minimax multiple shrinkage estimators. Ann. Statist. 14 188-205.
  • [14] Gradshtey n, I. and Ry zhik, I. (1994). Tables of Integrals, Series and Products, 5th ed. Academic Press, New York.
  • [15] Heath, D. and Sudderth, W. (1989). Coherent inference from improper priors and from finitely additive priors. Ann. Statist. 17 907-919.
  • [16] Lehmann, E. L. (1983). Theory of Point Estimation. Wiley, New York.
  • [17] O'Hagan, A. (1988). Modeling with heavy tails. In Bayesian Statistics 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.) 345-359. Oxford Univ. Press.
  • [18] Sans ´o, B. and Pericchi, L. R. (1992). Near ignorance classes of log-concave priors for the location model. Test 1 39-46.
  • [19] Stein, C. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 1135-1151.
  • [20] Strawderman, W. E. (1970). On the existence of proper Bay es minimax estimators of the mean of a multivariate normal distribution. Proc. Sixth Berkeley Sy mp. Math. Statist. Prob. 1 51-55. Univ. California Press, Berkeley.
  • [21] Strawderman, W. E. (1971). Proper bay es minimax estimators of the multivariate normal mean. Ann. Math. Statist. 42 385-388.
  • [22] Widder, D. V. (1946). The Laplace Transform. Princeton Univ. Press.