Advertisement

On Bayes procedures

  • Lorraine Schwartz
Article

Summary

A result of Doob regarding consistency of Bayes estimators is extended to a large class of Bayes decision procedures in which the loss functions are not necessarily convex. Rather weak conditions are given under which the Bayes procedures are consistent. One set involves restrictions on the a priori distribution and follows an example in which the choice of a priori distribution determines whether the Bayes estimators are consistent. Another example shows that the maximum likelihood estimators may be consistent when the Bayes estimators are not. However, the conditions given are of an essentially weaker nature than those established for consistency of maximum likelihood estimators.

Keywords

Stochastic Process Probability Theory Loss Function Large Class Mathematical Biology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bahadur, R. R.: Examples of the inconsistency of the maximum likelihood estimate, Sankhya, 20, 207–210 (1958).Google Scholar
  2. [2]
    Chernoff, H.: Sequential design of experiments. Ann. math. Statistics, 30, 755–770 (1959).Google Scholar
  3. [3]
    Doob, J. L.: Application of the theory of martingales. Colloque international Centre nat. Rech. Sci, Paris, 22–28 (1949).Google Scholar
  4. [4]
    Freedman, D. A.: On the asymptotic behaviour of Bayes estimates in the discrete case, Ann. math. Statistics, 34, 1386–1403 (1963).Google Scholar
  5. [5]
    Katz, M., and A. J. Thomasian: A bound for the law of large numbers for discrete Markov processes, Ann. math. Statistics 32, 336–337 (1961).Google Scholar
  6. [6]
    Kiefer, J., and J. Wolfowitz: Consistency of the maximum likelihood estimate in the presence of infinitely many incidental parameters, Ann. math. Statistics, 27, 887–906 (1956).Google Scholar
  7. [7]
    Kraft, C.: Some conditions for consistency and uniform consistency of statistical procedures, Univ. California, Publ. Statist. 2, 125–142 (1955).Google Scholar
  8. [8]
    LeCam, L.: On some asymptotic properties of the maximum likelihood estimates and related Bayes estimates. Univ. California Pub. Statist., 1, 277–330 (1953).Google Scholar
  9. [9]
    —: Les Proprietes Asymptotiques des Solutions de Bayes. Publ. Inst. Statist. Univ. Paris. 7, 17–35 (1958).Google Scholar
  10. [10]
    —, and L. Schwartz: A necessary and sufficient condition for the existence of consistent estimates, Ann. math. Statistics 31, 140–150 (1960).Google Scholar
  11. [11]
    Wald, A.: Note on the consistency of the maximum likelihood estimates, Ann. math. Statistics 20, 595–601 (1949).Google Scholar

Copyright information

© Springer-Verlag 1965

Authors and Affiliations

  • Lorraine Schwartz
    • 1
  1. 1.Dept. of MathematicsUniversity of British ColumbiaVancouver 8Canada

Personalised recommendations