Journal of Mathematical Sciences

, Volume 152, Issue 6, pp 869–874 | Cite as

On semiparametric statistical inferences in the moderate deviation zone

  • M. S. Ermakov


Recently, we have shown that the Hajek-Le Cam lower bound for asymptotic efficiency in estimation and the lower bound for the Pitman efficiency in hypothesis testing can be extended to the moderate deviation zone if weak additional assumptions hold. In this paper, we present a version of these results in the semiparametric form. Bibliography: 26 titles.


Russia Hypothesis Testing Standard Approach Statistical Inference Additional Assumption 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. R. Bahadur, “Asymptotic efficiency of tests and estimates,” Sankhya, 22, 229–252 (1960).MATHMathSciNetGoogle Scholar
  2. 2.
    P. Bickel, C. Klaassen, Y. Ritov, and J. A. Wellner, Efficient and Adaptive Estimation for the Semiparametric Models, John Hopkins Univ. Press, Baltimore (1993).Google Scholar
  3. 3.
    P. J. Bickel and J. Kwon, “Inference for semiparametric models: some questions and an answer,” Statistica Sinica, 11, 863–960 (1993).MathSciNetGoogle Scholar
  4. 4.
    A. A. Borovkov and A. A. Mogulskii, “Large deviations and testing of statistical hypothesis,” Trudy Sib. Otd. Inst. Mat. RAN (1992).Google Scholar
  5. 5.
    A. A. Borovkov and A. A. Mogulskii, “Large deviations and statistical invariance principle,” Teor. Veroyatn. Primen., 37, 11–18 (1992).MathSciNetGoogle Scholar
  6. 6.
    H. Chernoff, “A measure of asymptotic efficiency for tests of a hypothesis based on sums of observations.” Ann. Math. Statist., 23, 493–507 (1952).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    D. L. Donoho and R. C. Liu, “Geometrizing rates of convergence. I,” Technical Report 137, Dept. Statistics, Univ. California, Berkeley (1987).Google Scholar
  8. 8.
    M. S. Ermakov, “Asymptotic minimaxity of usual goodness of fit tests,” in: Grigelionis et al. (eds.), Proc. 5th Vilnius Conf. Probability Theory and Mathematical Statistics, (1991), pp. 323–331.Google Scholar
  9. 9.
    M. S. Ermakov, “On asymptotic minimaxity of rank tests,” Statist. Probab. Lett., 15, 191–196 (1992).MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    M. S. Ermakov, “Large deviations of empirical measures and hypothesis testing,” Zap. Nauchn. Semin. POMI, 207, 37–60 (1993).Google Scholar
  11. 11.
    M. S. Ermakov, “Asymptotically efficient statistical inference for moderate deviation probabilities,” Teor. Veroyatn. Primen., 48, 676–700 (2003).MathSciNetGoogle Scholar
  12. 12.
    M. S. Ermakov, “Importance sampling for simulation of moderate deviations of statistics,” Statistics Decision (2008) (to appear).Google Scholar
  13. 13.
    J. Hajek, “Local asymptotic minimax and admisibility in estimation,” in: Proc. Sixth Berkeley Symp. Math. Statist. Probab, California Univ. Press (1972), pp. 175–194.Google Scholar
  14. 14.
    J. L. Hodges and E. L. Lehmann, “On some nonparametric competitors of the t-test,” Ann. Math. Statist., 27, 324–335 (1956).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    I. A. Ibragimov and R. Z. Khasminskii, “Asymptotically normal families of distributions and efficient estimation,” Ann. Statist., 19, 1681–1721 (1991).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    W. C. M. Kallenberg, “Intermediate efficiency, theory, and examples,” Ann. Statist., 11, 170–182 (1983).MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Yu. A. Koshevnik and B. Ya. Levit, “On a nonparametric version of the information matrix,” Teor. Veroyatn. Primen., 21, 738–753 (1976).MATHGoogle Scholar
  18. 18.
    S. Kourauklis, “On the relation between Hodges-Lehman efficiency and Pitman efficiency,” Canad. J. Statist., 17, 311–318 (1989).CrossRefMathSciNetGoogle Scholar
  19. 19.
    S. Kourauklis, “A relation between the Chernoff index and the Pitman efficiency,” Statist. Probab. Lett., 9, 391–395 (1990).CrossRefMathSciNetGoogle Scholar
  20. 20.
    L. Le Cam, Asymptotic Methods in Statistical Decision Theory, Springer-Verlag (1986).Google Scholar
  21. 21.
    B. Ya. Levit, “On optimality of some statistical estimates,” in: Proc. Prague Sympos. Asymptotic Statist., 2 (1974), pp. 215–238.MathSciNetGoogle Scholar
  22. 22.
    M. Radavichius, “From asymptotic efficiency in minimax sense to Bahadur efficiency,” in: V. Sazonov and T. Shervashidze (eds), New Trends Probab. Statist, Vilnius (1991), pp. 629–635.Google Scholar
  23. 23.
    Ch. Stein, “Efficient nonparametric testing and estimation,” in: Third Berkeley Symp. Math. Statist. Probab., Univ. California Press (1956), pp. 187–195.Google Scholar
  24. 24.
    A. W. van der Vaart, “On differentiable functionals,” Ann. Statist., 19, 178–204 (1991).MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    A. W. van der Vaart, Asymptotic Statistics, Cambridge Univ. Press (1998).Google Scholar
  26. 26.
    H. S. Wieand, “A condition under which the Pitman and Bahadur approaches to efficiency coincide,” Ann. Statist., 4, 1003–1011 (1976).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.St.Petersburg State UniversitySt.PetersburgRussia

Personalised recommendations