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Journal of Mathematical Sciences

, Volume 88, Issue 6, pp 764–777 | Cite as

A method for obtaining asymptotic expansions under alternatives based on the properties of the likelihood ratio

  • V. E. Bening
Testing of Statistical Hypotheses

Keywords

Likelihood Ratio Asymptotic Expansion 
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References

  1. 1.
    W. Albers, P. J. Bickel, and W. R. Van Zwet, “Asymptotic expansion for the power of distribution free tests in the one-sample problem”,Ann. Math. Statist., No. 4, 108–156 (1976); No. 6, 1170–1171 (1978).Google Scholar
  2. 2.
    W. Albers,Asymptotic Expansions and the Deficiency Concept in Statistics, Mathematical Centre Tracts 58, Amsterdam (1974).Google Scholar
  3. 3.
    H. Bergstrôm and M. L. Puri, “Convergence and remainder terms in linear rank statistics”,Ann. Math. Statist., No. 5, 871–880 (1977).Google Scholar
  4. 4.
    P. J. Bickel, “Edgeworth expansions in nonparametric statistics”,Ann. Math. Statist., No. 2, 1–20 (1974).MATHMathSciNetGoogle Scholar
  5. 5.
    P. J. Bickel, D. M. Chibisov, and W. R. Van Zwet, “On efficiency of first and second order”,Statist. Rev., No., 49, 169–175 (1981).Google Scholar
  6. 6.
    P. J. Bickel and W. R. Van Zwet, “Asymptotic expansion for the power of distribution free tests in the one-sample problem”,Ann. Math. Statist., No. 6, 937–1004 (1978).Google Scholar
  7. 7.
    V. E. Bening, “Asymptotic expansions for Neyman'sC(α) tests,” inProceedings of the Third Vilnius Conference on Probability Theory and Mathematical Statistics [in Russian], Vilnius (1981), pp. 47–48.Google Scholar
  8. 8.
    V. E. Bening, “An asymptotic expansion for the distribution of a statistic admitting a stochastic decomposition depending on a linear combination of order statistics under local alternatives”,Sov. Mat. Dokl.,251, No. 1, 14–16 (1980).MATHMathSciNetGoogle Scholar
  9. 9.
    V. E. Bening, “Asymptotic expansions forL- andR-statistics under alternatives”,J. Math. Sci.,76, No. 2, 2227–2240 (1995).MathSciNetGoogle Scholar
  10. 10.
    V. E. Bening, “Asymptotic expansions under local alternatives”, in:Proceedings of the Fourth USSR-Japan Symposium on Probability Theory and Mathematical Statistics (1982), pp. 120–121.Google Scholar
  11. 11.
    D. M. Chibisov, “Asymptotic expansion for some asymptotically optimal test”, inProceedings of the Prague Symposium on Asymptotic Statistics, Charles Univ. Press, Prague (1974), pp. 37–68.Google Scholar
  12. 12.
    D. M. Chibisov, “Power and deficiency of asymptotically optimal tests”,Teor. Veroyatn. Primen.,30, 269–288 (1985).MATHMathSciNetGoogle Scholar
  13. 13.
    D. M. Chibisov, “Higher order properties of asymptotically optimal tests in a one-parametric family”, in:Proceedings of the Fourth USSR-Japan Symposium on Probability Theory and Mathematical Statistics, Tbilisi (1982), pp. 171–173.Google Scholar
  14. 14.
    D. M. Chibisov and W. R. Van Zwet, “On the Edgeworth expansion for the logarithm of the likelihood ratio”,Teor. Veroyatn. Primen.,29, No. 3, 417–439 (1984).Google Scholar
  15. 15.
    D. M. Chibisov, “Asymptotic expansion and deficiencies of tests”, in:Proceedings of the International Congress of Mathematicians, Warszawa (1983), pp. 1063–1079.Google Scholar
  16. 16.
    H. Cramér,Random Variables and Probability Distributions, Cambridge University Press, Cambridge (1937).Google Scholar
  17. 17.
    S. W. Dharmadhikari and K. Jogdeo, “Bounds on moments of certain random variables”,Ann. Math. Statist.,40, 1506–1508 (1969).MathSciNetGoogle Scholar
  18. 18.
    W. Feller,An Introduction to Probability Theory and Its Applicaions, Vols. 1, 2, Wiley, New York (1968), (1971).Google Scholar
  19. 19.
    L. Le Cam,Locally Asymptotically Normal Families of Distributions Univ. of California Press (1969), pp. 37–98.Google Scholar
  20. 20.
    M. Loeve,Probability Theory (1955).Google Scholar
  21. 21.
    S. V. Nagaev, “Some limit theorens for large deviations,”Teor. Veroyatn. Primen.,10, No. 2, 231–254 (1965).MATHMathSciNetGoogle Scholar
  22. 22.
    J. Pfanzagl, “Asymptotically optimum estimation and test procedures,” in:Proceedings of the Prague Symposium on Asymptotic Statistics, Prague (1974), pp. 201–272.Google Scholar
  23. 23.
    J. Pfanzagl, “First order efficiency implies second order efficiency,” inContributions to Statistics, Jaroslaw Hajek Memorial Volume, Prague (1979), pp. 167–196.Google Scholar
  24. 24.
    J. Pfanzagl, “Asymptotic expansions in parametric statistical theory,” in:Developments in Statistics, New York (1980), pp. 1–97.Google Scholar
  25. 25.
    J. Hajek and Z. Sidak,Theory of Rank Tests, Prague (1967).Google Scholar
  26. 26.
    R. Helmers, “A Berry-Esseen theorem for linear combinations of order statistics,”Ann. Probab., No. 9, 342–347 (1981).MATHMathSciNetGoogle Scholar
  27. 27.
    R. Helmers, “Edgeworth expantions for linear combinations of order statistics with smooth weight functions,”Ann. Statist., No. 8, 1361–1374 (1980).MATHMathSciNetGoogle Scholar
  28. 28.
    J. L. Hodges Jr. and E. L. Lehmann, “Deficiency”,Ann. Math. Statist., 783–801 (1970).Google Scholar
  29. 29.
    W. R. Van Zwet, “Asymptotic expansions for the distribution function of linear combinations of order statistics,” in:Statistical Decision Theory and Related Topics, Vol. 2, New York (1977), pp. 421–438.Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • V. E. Bening
    • 1
  1. 1.MoscowRussia

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