Advertisement

Locally averaged risk

  • Khursheed Alam
  • James R. Thompson
Article

Summary

A heuristic method of reducing a class of admissible or Bayes decision rules is given. A new risk function is defined which is called the locally averaged risk. Bayes and admissible rules with respect to the new risk function are calledG-Bayes andG-admissible, respectively. It is shown under general assumptions that the class ofG-Bayes decision rules is a subset of the class of Bayes decision rules and the class ofG-admissible decision rules is a subset of the class of admissible decision rules.

Some examples are considered, showing that the usual estimates of the parameter of a distribution with squared error as loss function, which are known to be admissible, are alsoG-admissible.

Keywords

Decision Rule Loss Function Prior Distribution Risk Function Usual Estimate 
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]
    T. S. Fergusan,Mathematical Statistics, A Theoretic Approach, Academic Press, New York and London, 1967.Google Scholar
  2. [2]
    M. A. Girshick and L. J. Savage, “Bayes and minimax estimates for quadratic loss functions,”Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley and Los Angeles, (1951), 53–74.Google Scholar
  3. [3]
    J. L. Hodges and E. L. Lehmann, “Some applications of the Cramer-Rao Inequality,”Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley and Los Angeles, (1951), 13–22.Google Scholar
  4. [4]
    S. Karlin, “Admissibility for estimation with quadratic loss,”Ann. Math. Statist. 29 (1958), 406–436.MathSciNetGoogle Scholar
  5. [5]
    E. J. G. Pitman, “The estimation of location and scale parameters of a continuous population,”Biometrika, 30 (1939), 391–421.Google Scholar
  6. [6]
    C. Stein, “The admissibility of Pitman’s estimator of a single location parameter,”Ann. Math. Statist., 30 (1959), 970–979.MathSciNetGoogle Scholar
  7. [7]
    A. Wald,Statistical Decision Functions, John Wiley, New York, 1958.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 1969

Authors and Affiliations

  • Khursheed Alam
    • 1
  • James R. Thompson
    • 1
  1. 1.Indiana UniversityUSA

Personalised recommendations