Advertisement

Approximation of Improper Prior Measures by Prior Probability Measures

  • Charles Stein

Abstract

It is known that, ordinarily, any admissible decision procedure for a statistical decision problem is, in a fairly strong sense, a limit of Bayes procedures, and in many cases such a limit must be a formal Bayes procedure with respect to a prior measure which may be improper (unbounded). This paper is, for the most part, a non-rigorous attempt at finding, in a reasonably explicit form, conditions for such a formal Bayes procedure to be admissible. In addition, a small amount of effort is devoted to the question of approximation of improper prior measures by prior probability measures without regard to the question of admissibility.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    James, W., and C. Stein: Estimation with quadratic loss. Proc. Fourth Berkeley Symposium on Math. Stat. and Prob. p. 361. Berkeley and Los Angeles: Univ. of Calif. Press 1961.Google Scholar
  2. [2]
    Jeffreys, H.: Theory of Probability. Third Edition. Oxford: Clarendon Press 1961.MATHGoogle Scholar
  3. [3]
    Kakutani, S.: On equivalence of infinite product measures. Ann. Math. 49, 214 (1948).MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Kiefer, J.: Invariance, minimax sequential estimation, and continuous time processes. Ann. Math. 28, 573 (1957).MathSciNetMATHGoogle Scholar
  5. [5]
    Kraft, C.: Some conditions for consistency and uniform consistency of statistical procedures. Univ. Calif. Publ. in Statistics. 2, No. 6, 125 (1955).MathSciNetGoogle Scholar
  6. [6]
    Kuno, H.: On minimax invariant estimators of the transformation parameter. Nat. Sci. Rep., Ochanomizu Univ. 6, 31 (1955).MathSciNetGoogle Scholar
  7. [7]
    Lehmann, E.: Testing Statistical Hypotheses. New York: Wiley 1959.MATHGoogle Scholar
  8. [8]
    Loomis, L.: An Introduction to Abstract Harmonic Analysis. New York: Van Nostrand 1953.MATHGoogle Scholar
  9. [9]
    Meyers, N., and J. Serrin: The exterior Dirichlet problem for second order partial differential equations. J. Math. Mech. 9, 513 (1960).MathSciNetMATHGoogle Scholar
  10. [10]
    Peisakoff, M.: Transformation of parameters, Unpublished Ph. D. Thesis, Princeton 1951.Google Scholar
  11. [11]
    Pitman, E. J. G.: Location and scale parameters. Biometrika 30, 391 (1939).MATHGoogle Scholar
  12. [12]
    Stein, C.: A necessary and sufficient condition for admissibility. Ann. Math. Statist. 26, 518 (1955).MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Stone, M.: The posterior t distribution. Ann. Math. Statist. 34, 568 (1963).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1965

Authors and Affiliations

  • Charles Stein
    • 1
  1. 1.Stanford UniversityUSA

Personalised recommendations