Journal of Mathematical Sciences

, Volume 204, Issue 1, pp 42–54 | Cite as

Exponentiality Tests Based on Ahsanullah’s Characterization and Their Efficiency

  • K. Yu. Volkova
  • Ya. Yu. Nikitin

We construct integral and supremum type tests of exponentiality based on Ahsanullah’s characterization of the exponential law. We discuss limiting distributions and large deviations of new test statistics under the null-hypothesis and calculate their local Bahadur efficiency under common parametric alternatives. Conditions of local optimality of the new statistics are given. Bibliography: 33 titles.


Kolmogorov Type Residual Life Function Leibler Information Bahadur Efficiency Exact Slope 
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  1. 1.
    V. V. Litvinova, Asymptotic properties of symmetry and goodness-of-fit texts based on characterizations, Candidate dissertation, SPbGU (2004).Google Scholar
  2. 2.
    A. V. Tchirina, Asymptotic efficiency of exponentiality tests that are free of scale parameter, Candidate dissertation, SPbGU (2006).Google Scholar
  3. 3.
    I. Ahmad and I. Alwasel, “A goodness-of-fit test for exponentiality based on the memoryless property,” J. Roy. Statist. Soc., 61, 681–689 (1999).CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    M. Ahsanullah, “On a characterization of the exponential distribution by spacings,” Ann. Inst. Statist. Math., 30, 163–166 (1978).CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    M. Z. Anis and S. Dutta, “Recent tests of exponentiality against IFR alternatives: a survey,” J. Statist. Comput. Simul., 80, 1373–1387 (2010).CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    J. E. Angus, “Goodness-of-fit tests for exponentiality based on a loss-of-memory type functional equation,” J. Statist. Plann. Infer., 6, 241–251 (1982).CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    S. Asher, “A survey of tests for exponentiality,” Commun. Statist. Theory Meth., 19, 1811–1825 (1990).CrossRefGoogle Scholar
  8. 8.
    R. R. Bahadur, Some Limit Theorems in Statistics, SIAM, Philadelphia (1971).CrossRefMATHGoogle Scholar
  9. 9.
    N. Balakrishnan and A. Basu, The Exponential Distribution: Theory, Methods and Applications, Gordon and Breach, Langhorne, PA (1995).Google Scholar
  10. 10.
    L. Baringhaus and N. Henze, “Tests of fit for exponentiality based on a characterization via the mean residual life function,” Statist. Papers, 41, 225–236 (2000).CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    A. DasGupta, Asymptotic Theory of Statistics and Probability, Springer, New York (2008).MATHGoogle Scholar
  12. 12.
    K. A. Doksum and B. S. Yandell, “Tests of exponentiality,” Handbook of Statist., 4, 579–612 (2000).CrossRefMathSciNetGoogle Scholar
  13. 13.
    A. Grane and J. Fortiana, “A location- and scale-free goodness-of-fit statistic for the exponential distribution based on maximum correlations,” Statist., 43, 1–12 (2009).CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    R. Helmers, P. Janssen, and R. Serfling, “Glivenko–Cantelli properties of some generalized empirical DF’s and strong convergence of generalized L-statistics,” Probab. Theory Relat. Fields, 79, 75–93 (2009).CrossRefMathSciNetGoogle Scholar
  15. 15.
    N. Henze and S. Meintanis, “Goodness-of-fit tests based on a new characterization of the exponential distribution,” Commun. Statist. Theor. Meth., 31, 1479–1497 (2002).CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    W. Hoeffding, “A class of statistics with asymptotically normal distribution,” Ann. Math. Statist., 19, 293–325 (1948).CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    P. L. Janssen, Generalized Empirical Distribution Functions with Statistical Applications, Limburgs Univ. Centrum, Diepenbeek (1988).Google Scholar
  18. 18.
    V. S. Korolyuk and Yu. V. Borovskikh, Theory of U-statistics, Kluwer, Dordrecht (1994).CrossRefMATHGoogle Scholar
  19. 19.
    H. L. Koul, “A test for new better than used,” Commun. Statist. Theory Meth., 6, 563–574 (1977).CrossRefMathSciNetGoogle Scholar
  20. 20.
    H. L. Koul, “Testing for new is better than used in expectation,” Commun. Statist. Theory Meth., 7, 685–701 (1978).CrossRefMathSciNetGoogle Scholar
  21. 21.
    C. D. Lai and M. Xie, Stochastic Ageing and Dependence for Reliability, Springer-Verlag, New York (2006).MATHGoogle Scholar
  22. 22.
    P. Nabendu, J. Chun, and R. Crouse, Handbook of Exponential and Related Distributions for Engineers and Scientists, Chapman and Hall (2002).Google Scholar
  23. 23.
    Y. Nikitin, Asymptotic Efficiency of Nonparametric Tests, Cambridge Univ. Press, New York (1995).CrossRefMATHGoogle Scholar
  24. 24.
    Ya. Yu. Nikitin, “Large deviations of U-empirical Kolmogorov–Smirnov tests, and their efficiency,” J. Nonpar. Statist., 22, 649–668 (2010).CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Ya. Yu. Nikitin and I. Peaucelle, “Efficiency and local optimality of distribution-free tests based on U- and V -statistics,” Metron, LXII, 185–200 (2004).MathSciNetGoogle Scholar
  26. 26.
    Ya. Yu. Nikitin and E. V. Ponikarov, “Rough large deviation asymptotics of Chernoff type for von Mises functionals and U-statistics,” Proc. St.Petersburg Math. Society, 7, 124–167 (1999).MathSciNetGoogle Scholar
  27. 27.
    Ya. Yu. Nikitin and A. V. Tchirina, “ Bahadur efficiency and local optimality of a test for the exponential distribution based on the Gini statistic,” Statist. Meth. Appl., 5, 163–175 (1996).MATHGoogle Scholar
  28. 28.
    Ya. Yu. Nikitin and A. V. Tchirina, “Lilliefors test for exponentiality: large deviations, asymptotic efficiency, and conditions of local optimality,” Math. Meth. Statist., 16, 16–24 (2007).CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Ya. Yu. Nikitin and K. Yu. Volkova, “Asymptotic efficiency of exponentiality tests based on order statistics characterization,” Georgian Math. J., 17, 749–763 (2010).MATHMathSciNetGoogle Scholar
  30. 30.
    R. F. Rank, Statistische Anpassungstests und Wahrscheinlichkeiten grosser Abweichungen, Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation, Hannover (1999).Google Scholar
  31. 31.
    J. S. Rao and E. Taufer, “The use of mean residual life to test departures from exponentiality,” J. Nonparam. Statist., 18, 277–292 (2006).CrossRefMATHGoogle Scholar
  32. 32.
    B. W. Silverman, “Convergence of a class of empirical distribution functions of dependent random variables,” Ann. Probab., 11, 745–751 (1983).CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    H. S. Wieand, “A condition under which the Pitman and Bahadur approaches to efficiency coincide,” Ann. Statist., 4, 1003–1011 (1976).CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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