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Some contributions to the asymptotic theory of Bayes solutions

  • P. J. Bickel
  • J. A. Yahav
Article

Summary

This paper deals with the asymptotic theory of Bayes solutions in (i) Estimation (ii) Testing when hypothesis and alternative are separated at least by an indifference region, under the assumption that the observations are independent and indentically distributed. The estimation results which are partial generalizations of results of LeCam begin with a proof of the convergence of the normalized posterior density to the appropriate normal density in a strong sense. From this result we derive the asymptotic efficiency of Bayes estimates obtained from smooth loss functions and in particular of the posterior mean. The last two theorems of this section deal with asymptotic expansions for the posterior risk in such estimation problems. The section on testing contains a limit theorem for the n-th root of the posterior risk under weak conditions on the prior and the loss function. Finally we discuss generalizations and some open problems.

Keywords

Open Problem Asymptotic Expansion Limit Theorem Loss Function Estimation Result 
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References

  1. 1.
    Bahadur, R. R.: An optimal property of the likelihood ratio statistic. Proc. fifth Berkeley Sympos. math. Statist. Probability (1965).Google Scholar
  2. 2.
    -, and P. J. Bickel: An optimal property of Bayes' test statistics. To be submitted to Sankhyā (1966).Google Scholar
  3. 3.
    Bickel, P. J., and J. A. Yahav: Asymptotically pointwise optimal procedures in sequential analysis. Proc. fifth Berkeley Sympos. math. Statist. Probability (1965).Google Scholar
  4. 4.
    — —: Asymptotically optimal procedures in Bayes and minimax sequential estimation. Ann. math. Statistics 39, 442–456 (1968).Google Scholar
  5. 5.
    Cramèr, H.: Methods of Mathematical Statics. Princeton Univ. Press 1946.Google Scholar
  6. 6.
    Davis, R. C.: Asymptotic properties of Bayes estimate. Ann. math. Statistics 22, 8, 484 (Abstract) (1951).Google Scholar
  7. 7.
    Dieudonné, J.: Foundations of Modern Analysis. New York: Interscience 1961.Google Scholar
  8. 8.
    Dunford, N., and J. Schwartz: Linear Operators Part I. New York: Interscience 1957.Google Scholar
  9. 9.
    Farrell, R.: Weak limits of sequences of Bayes procedures in estimation theory. Proc. fifth Berkeley Sympos. math. Statist. Probability (1965).Google Scholar
  10. 10.
    Kiefer, J., and J. Wolfowitz: Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann. math. Statistics 27, 887–906 (1956).Google Scholar
  11. 11.
    —, and J. Sacks: Asymptotically optimum sequential inference and design. Ann. math. Statistics 34, 705–750 (1963).Google Scholar
  12. 12.
    Le Cam, L.: On some asymptotic properties of maximum likelihood estimates and related Bayes estimates. Univ. California Publ. Statist. 1, 277–330 (1953).Google Scholar
  13. 13.
    - On the asymptotic theory of estimation and testing hypotheses. Proc. third Berkeley Symp. math. Statist. Probability, 129–157 (1955).Google Scholar
  14. 14.
    —: Les propriétés asymptotiques des solutions de Bayes. Publ. Inst. Statist. Univ. Paris 7, 18–35 (1958).Google Scholar
  15. 15.
    Rubin, H., and J. Sethuraman: Probabilities of moderate deviations. Sankhyā 27, 325–346 (1965).Google Scholar
  16. 16.
    Wald, A.: A note on the consistency of the maximum likelihood estimate. Ann. math. Statistics 20, 595–601 (1949).Google Scholar
  17. 17.
    Wolfowitz, J.: Method of maximum likelihood and the Wald theory of decision functions. Indagationes math. 15, 114–119 (1953).Google Scholar
  18. 18.
    Jeffreys, H.: Theory of probability, 3rd Ed. Oxford Univ. Press 1961.Google Scholar
  19. 19.
    Gomberg, D.: Doctoral dissertation. Univ. of California, Berkeley 1967.Google Scholar
  20. 20.
    Schwartz, L.: On Bayes procedures. Z. Wahrscheinlichkeitstheorie verw. Geb. 4, 10–26 (1965).Google Scholar
  21. 21.
    Elfving, G.: Robustness of Bayes decisions against choice of prior. Tech. Report 122, Stanford University 1966.Google Scholar
  22. 22.
    Bickel, P. J., and J. A. Yahav: On an asymptotically optimal rule in sequential estimation with quadratic loss. To appear in Ann. Math. Statistics, April 1969.Google Scholar

Copyright information

© Springer-Verlag 1969

Authors and Affiliations

  • P. J. Bickel
    • 1
  • J. A. Yahav
    • 2
  1. 1.Dept. of StatisticsUniversity of CaliforniaBerkeleyUSA
  2. 2.Dept. of Mathem. StatisticsUniversity of Tel AvivTel AvivIsrael

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