The Annals of Statistics

Asymptotically optimal and admissible decision rules in compound compact Gaussian shift experiments

Suman Majumdar

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Abstract

Asymptotically optimal and admissible compound decision rules are obtained in a Hilbert-parameterized Gaussian shift experiment. The component parameter set is restricted to compact. For the squared error loss, every compound Bayes estimator is admissible and every compound estimator Bayes versus full support hyperprior mixture of iid priors on the compound parameter is asymptotically optimal. For the latter class of rules induced by full support hyperpriors, asymptotic optimality and admissibility extend to equi- (in decisions) uniformly continuous and bounded risk functions. Normality of certain mixtures of the standard Gaussian process and qualitative robustness of the component Bayes estimator (results of independent interest used in the paper) are derived.

Article information

Source
Ann. Statist., Volume 24, Number 1 (1996), 196-211.

Dates
First available in Project Euclid: 26 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1033066206

Digital Object Identifier
doi:10.1214/aos/1033066206

Mathematical Reviews number (MathSciNet)
MR1389887

Zentralblatt MATH identifier
0853.62012

Subjects
Primary: 62C25: Compound decision problems
Secondary: 62C15: Admissibility 62E15: Exact distribution theory 62F35: Robustness and adaptive procedures

Keywords
Compound Bayes rules component Bayes estimator asymptotic optimality admissibility full support hyperprior consistency Gaussian shift experiment qualitative robustness

Citation

Majumdar, Suman. Asymptotically optimal and admissible decision rules in compound compact Gaussian shift experiments. Ann. Statist. 24 (1996), no. 1, 196--211. doi:10.1214/aos/1033066206. https://projecteuclid.org/euclid.aos/1033066206


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References

  • DATTA, S. 1988. Asy mptotically optimal Bay es compound and empirical Bay es estimators in exponential families with compact parameter space. Ph.D. dissertation, Dept. Statistics and Probability, Michigan State Univ. Z.
  • DATTA, S. 1991. Asy mptotic optimality of Bay es compound estimators in compact exponential families. Ann. Statist. 19 354 365. Z.
  • DUDLEY, R. M. 1989. Real Analy sis and Probability. Wadsworth & Brooks Cole, Belmont, CA. Z.
  • FERGUSON, T. S. 1967. Mathematical Statistics, A Decision Theoretic Approach. Academic Press, New York. Z.
  • GILLILAND, D. C. and HANNAN, J. 1986. The finite state compound decision problem, equivariance and restricted risk components. In Adaptive Statistical Procedures and Related Z. Topics J. Van Ry zin, ed. 129 145. IMS, Hay ward, CA. Z.
  • GILLILAND, D. C., HANNAN, J. and HUANG, J. S. 1976. Asy mptotic solutions to the two state component compound decision problem, Bay es versus diffuse priors on proportions. Ann. Statist. 4 1101 1112. Z.
  • HANNAN, J. F. and ROBBINS, H. 1955. Asy mptotic solutions of the compound decision problem for two completely specified distributions. Ann. Math. Statist. 36 1743 1752. Z.
  • IBRAGIMOV, I. A. and KHAS'MINSKII, R. Z. 1991. Asy mptotically normal families of distributions and efficient estimation. Ann. Statist. 19 1681 1724. Z.
  • LE CAM, L. M. 1986. Asy mptotic Methods in Statistical Decision Theory. Springer, New York. Z.
  • LE CAM, L. M. and YANG, G. L. 1990. Asy mptotics in Statistics, Some Basic Concepts. Springer, New York. Z.
  • MAJUMDAR, S. 1993. Posterior consistency in Gaussian shift experiments. Technical Report 247, Dept. Statistics and Applied Probability, Univ. California, Santa Barbara. Z.
  • MAJUMDAR, S. 1994. Bay es compound and empirical Bay es estimation of the mean of a Gaussian distribution on a Hilbert space. J. Multivariate Anal. 48 87 106. Z.
  • MAJUMDAR, S., GILLILAND, D. and HANNAN, J. 1993. Bounds for robust ML and posterior consistency in compound mixture state experiments. Unpublished manuscript. Z.
  • MASHAy EKHI, M. 1990. Stability of sy mmetrized probabilities and compact compound equivariant decisions. Ph.D. dissertation, Dept. Statistics and Probability, Michigan State Univ. Z.
  • MASHAy EKHI, M. 1993. On equivariance and the compound decision problem. Ann. Statist. 21 736 745. Z.
  • MILLAR, P. W. 1983. The Minimax Principle in Asy mptotic Statistical Theory. Lecture Notes in Math. 976. Springer, New York.
  • PARTHASARATHY, K. R. 1967. Probability Measures on Metric Spaces. Academic Press, New York. Z.
  • ROBBINS, H. 1951. Asy mptotically subminimax solutions of compound statistical decision problems. Proc. Second Berkeley Sy mp. Math. Statist. Probab. 131 148. Univ. California Press, Berkeley. Z.
  • RUDIN, W. 1973. Functional Analy sis. McGraw-Hill, New York. Z.
  • RUDIN, W. 1987. Real and Complex Analy sis. McGraw-Hill, Singapore. Z.
  • STRASSER, H. 1985. Mathematical Theory of Statistics: Statistical Experiments and Asy mptotic Decision Theory. de Gruy ter, Berlin. Z.
  • ZHU, J. 1992. Asy mptotic behavior of compound rules in compact families. Unpublished manuscript.
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