Advertisement

Goodness-of-Fit Tests Based on Characterization of Uniformity by the Ratio of Order Statistics, and Their Efficiency

  • 1 Accesses

We construct integral and supremum type goodness-of-fit tests for the uniform law based on Ahsanullah’s characterization of the uniform law. We discuss limiting distributions of new tests and describe the logarithmic large deviation asymptotics of test statistics under null-hypothesis. This enables us to calculate their local Bahadur efficiency under some parametric alternatives. Conditions of local optimality of new statistics are given.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.

References

  1. 1.

    M. Ahsanullah, “On characterizations of the uniform distribution based on functions of order statistics,” Aligarh J. Statist., 9, 1–6 (1989).

  2. 2.

    R. R. Bahadur, Some Limit Theorems in Statistics, SIAM, Philadelphia (1971).

  3. 3.

    R. R. Bahadur, “Stochastic comparison of tests,” Ann. Math. Statist., 31, 276–295 (1960).

  4. 4.

    R. R. Bahadur, “Rates of convergence of estimates and test statistics,” Ann. Math. Statist., 38, 303–324 (1967).

  5. 5.

    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).

  6. 6.

    A. DasGupta, Asymptotic Theory of Statistics and Probability, Springer, New York (2008).

  7. 7.

    T. Hashimoto and S. Shirahata, “A goodness-of-fit test based on a characterization of uniform distribution,” J. Japan Statist. Soc., 23, 123–130 (1993).

  8. 8.

    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).

  9. 9.

    R. Helmers, P. Janssen, and R. Serfling, “Glivenko–Cantelli properties of some generalized empirical DF’s and strong convergence of generalized L-statistics,” Probab. Theor. Relat. Fields, 79, 75–93 (1988).

  10. 10.

    M. Jovanović, B. Milošević, Ya. Yu. Nikitin, M. Obradović, and K. Yu. Volkova, “Tests of exponentiality based on Arnold–Villasenor characterization and their efficiencies,” Computat. Statist. Data Anal., 90, 100–113 (2015).

  11. 11.

    V. S. Korolyuk and Y. V. Borovskikh, Theory of U-statistics, 2nd ed., Springer Science and Business Media (2013).

  12. 12.

    B. Milošević, “Asymptotic efficiency of goodness-of-fit tests based on Too–Lin characterization,” ArXiv:1508.05314 (2015).

  13. 13.

    B. Milošević, “Asymptotic efficiency of new exponentiality tests based on a characterization,” Metrika, 79, 221–236 (2016).

  14. 14.

    K. Morris and D. Szynal, “Goodness-of-fit tests using characterizations of continuous distributions,” Appl. Math. (Warsaw), 28, 151–168 (2001).

  15. 15.

    Y. Nikitin, Asymptotic Efficiency of Nonparametric Tests, Cambridge Univ. Press, New York (1995).

  16. 16.

    Ya. Yu. Nikitin, “Large deviations of U-empirical Kolmogorov–Smirnov tests, and their efficiency,” J. Nonpar. Statist.,22, 649–668 (2010).

  17. 17.

    Ya. Yu. Nikitin, “Tests based on characterizations, and their efficiencies,” Acta et Comment. Univ. Tartuens. Matem., 21, 3–24 (2017).

  18. 18.

    P. Muliere and Y. Nikitin, “Scale-invariant test of normality based on Polya’s characterization,” Metron, 60, 21–33 (2002).

  19. 19.

    Ya. Yu. Nikitin and I. Peaucelle, “Efficiency and local optimality of distribution-free tests based on U and V -statistics,” Metron, 62, 185–200 (2004).

  20. 20.

    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. Soc., 7, 124–167 (1999).

  21. 21.

    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).

  22. 22.

    Ya. Yu. Nikitin and K. Yu. Volkova, “Asymptotic efficiency of exponentiality tests based on order statistics characterization,” Georgian Math. J., 17, 749–763 (2010).

  23. 23.

    M. Obradović, “Three characterizations of exponential distribution involving the median of sample of size three,” J. Statist. Theory Appl., 14, 257–264 (2015).

  24. 24.

    M. Obradović, M. Jovanovic, and B. Milošević, “Goodness-of-fit tests for Pareto distribution based on a characterization and their asymptotics,” Statistics, 49, 1026–1041 (2015).

  25. 25.

    B. W. Silverman, “Convergence of a class of empirical distribution functions of dependent random variables,” Ann. Probab., 11, 745–751 (1983).

Download references

Author information

Correspondence to K. Yu. Volkova.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 466, 2017, pp. 67–80.

Translated by S. Yu. Pilyugin.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Volkova, K.Y., Karakulov, M.S. & Nikitin, Y.Y. Goodness-of-Fit Tests Based on Characterization of Uniformity by the Ratio of Order Statistics, and Their Efficiency. J Math Sci 244, 743–751 (2020) doi:10.1007/s10958-020-04647-x

Download citation