On the asymptotic defect of some Bayesian criteria*

This paper focuses on the problem of testing a simple hypothesis about a one-dimensional parameter against one-sided alternatives with independent identically distributed random variables. A formula for extreme deviation of the power function from the envelope power function is obtained for asymptotically efficient Bayesian criteria based on a Bayesian likelihood ratio. This formula makes it possible to find the asymptotic deficiency in terms of the Hodges–Lehmann deficiency. The method used here makes it possible to relax necessary regularity conditions.

This is a preview of subscription content, access via your institution.


  1. 1.

    D. M. Chibisov, “Power and deficiency of asymptotically optimal tests,” Teor. Veroyatn. Primen., 30, No. 2, 269–288 (1985).

    MathSciNet  Google Scholar 

  2. 2.

    S. I. Gusev, “Asymptotic expansions associated with some statistical estimates in the smooth case,” Teor. Veroyatn. Primen., 20, No. 3, 488–514 (1975).

    MathSciNet  Google Scholar 

  3. 3.

    I. A. Ibragimov and R. Z. Hasminskii, Asymptotic Estimation Theory, Nauka, Moscow (1979).

    Google Scholar 

  4. 4.

    D. M. Chibisov and W. R. Van Zwet, “On the Edgeworth expansion for the logarithm of the likelihood ratio,” Theor. Probab. Appl., 29, No. 3, 417–543 (1984).

    MATH  Google Scholar 

  5. 5.

    V. E. Bening and D. M. Chibisov, “Higher-order asymptotic optimality in testing problems with nuisance parameters,” Math. Methods Stat., 8, No. 2, 142–165 (1999).

    MathSciNet  MATH  Google Scholar 

  6. 6.

    V. E. Bening, “A formula for deficiency: one sample L- and R-tests,” Math. Meth. Stat., 5, No. 2, 3, 167–188; 274–293 (1995).

    MathSciNet  Google Scholar 

  7. 7.

    J. Pfanzagl, “Asymptotic expansions in parametric statistical theory,” in: Development in Statistics, Vol.3, Academic Press, New York (1980), 1–97.

  8. 8.

    D. M. Chibisov, “Asymptotic expansions and deficiencies of tests,” in: Proc. Intern. Congress of Mathematicians, Vol. 2, Warszawa (1983), 1063–1079.

  9. 9.

    S. V. Nagaev, “Some limit theorems for large deviations,” Teor. Veroyatn. Primen., 10, No. 2, 231–254 (1965).

    MathSciNet  Google Scholar 

  10. 10.

    J. Pfanzagl, “On asymptotically complete classes,” Statistical Inference and Related Topics, Vol.7, Academic Press, New York (1975), 1–43.

  11. 11.

    M. G. Kendall and A. Stuart, Distribution theory, Griffin and Co. (2000).

  12. 12.

    D. M. Chibisov, “Asymptotic expansions for Neyman’s C(α) tests,” in: Proc. 2nd Japan-USSR Sypmos. Prob. Theory, Lecture Notes in Mathematics, Vol.330, Springer (1973), 16–45.

Download references

Author information



Corresponding author

Correspondence to V. E. Bening.

Additional information

*Research supported by the Russian Foundation for Basic Research, projects No. 02–01–00949.

Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, Vol. 17, pp. 11–23, 2003.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bening, V.E. On the asymptotic defect of some Bayesian criteria*. J Math Sci 189, 967–975 (2013). https://doi.org/10.1007/s10958-013-1239-3

Download citation


  • Asymptotic Expansion
  • Regularity Condition
  • Nuisance Parameter
  • Extreme Deviation
  • Simple Hypothesis