Advertisement

Metrika

, Volume 79, Issue 2, pp 221–236 | Cite as

Asymptotic efficiency of new exponentiality tests based on a characterization

  • Bojana MiloševićEmail author
Article

Abstract

Two new tests for exponentiality, of integral- and Kolmogorov-type, are proposed. They are based on a recent characterization and formed using appropriate V-statistics. Their asymptotic properties are examined and their local Bahadur efficiencies against some common alternatives are found. A class of locally optimal alternatives for each test is obtained. The powers of these tests, for some small sample sizes, are compared with different exponentiality tests.

Keywords

Testing of exponentiality Order statistics Bahadur efficiency U-statistics 

Mathematics Subject Classification

60F10 62G10 62G20 62G30 

Notes

Acknowledgments

We would like to thank the Editor and the Referee for their very useful remarks.

References

  1. Ahmad I, Alwasel I (1999) A goodness-of-fit test for exponentiality based on the memoryless property. J R Stat Soc Ser B Stat Methodol 61(3):681–689CrossRefMathSciNetzbMATHGoogle Scholar
  2. Ahsanullah M, Hamedani GG (2010) Exponential distribution: theory and methods. NOVA Science, New YorkGoogle Scholar
  3. Angus JE (1982) Goodness-of-fit tests for exponentiality based on a loss-of-memory type functional equation. J Stat Plan Inference 6(3):241–251CrossRefMathSciNetzbMATHGoogle Scholar
  4. Arnold BC, Balakrishnan N, Nagaraja HN (2008) A first course in order statistics. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  5. Bahadur RR (1971) Some limit theorems in statistics. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  6. Balakrishnan N, Rao CR (1998) Order statistics, theory and methods. Elsevier, AmsterdamGoogle Scholar
  7. Galambos J, Kotz S (1978) Characterizations of probability distributions. Springer, BerlinzbMATHGoogle Scholar
  8. Gomez YM, Bolfarine H, Gomez HW (2014) A new extension of the exponential distribution. Rev C Estad 37(1):25–34CrossRefMathSciNetGoogle Scholar
  9. Helmers R, Janssen P, Serfling R (1988) Glivenko–Cantelli properties of some generalized empirical DF’s and strong convergence of generalized L-statistics. Probab Theory Rel Fields 79(1):75–93CrossRefMathSciNetzbMATHGoogle Scholar
  10. Henze N, Meintanis SG (2005) Recent and clasicical tests for exponentiality: a partial review with comparisons. Metrika 61(1):29–45CrossRefMathSciNetzbMATHGoogle Scholar
  11. Hoeffding W (1948) A class of statistics with asymptotically normal distribution. Ann Math Stat 19:293–325CrossRefMathSciNetzbMATHGoogle Scholar
  12. Jevremović V (1991) A note on mixed exponential distribution with negative weights. Stat Probab Lett 11(3):259–265CrossRefzbMATHGoogle Scholar
  13. Korolyuk VS, Borovskikh YV (1994) Theory of \(U\)-statistics. Kluwer, DordrechtCrossRefGoogle Scholar
  14. Koul HL (1977) A test for new better than used. Commun Stat Theory Method 6(6):563–574CrossRefMathSciNetGoogle Scholar
  15. Koul HL (1978) Testing for new is better than used in expectation. Commun Stat Theory Method 7(7):685–701Google Scholar
  16. Nadarajah S, Haghighi F (2010) An extension of the exponential distribution. Statistics 45(6):543–558CrossRefMathSciNetGoogle Scholar
  17. Nikitin Y (1995) Asymptotic efficiency of nonparametric tests. Cambridge University Press, New YorkCrossRefzbMATHGoogle Scholar
  18. Nikitin YY (1996) Bahadur efficiency of a test of exponentiality based on a loss of memory type functional equation. J Nonparametr Stat 6(1):13–26CrossRefzbMATHGoogle Scholar
  19. Nikitin Y, Peaucelle I (2004) Efficiency and local optimality of distribution-free tests based on U- and V-statistics. Metron 62(2):185–200MathSciNetGoogle Scholar
  20. Nikitin YY (2010) Large deviations of \(U\)-empirical Kolmogorov-Smirnov tests, and their efficiency. J Nonparametr Stat 22(5):649–668Google Scholar
  21. Nikitin YY (1984) Local asymptotic bahadur optimality and characterization problems. Theory Probab Appl 29:79–92CrossRefzbMATHGoogle Scholar
  22. Nikitin YY, Ponikarov EV (1999) Rough large deviation asymptotics of Chernoff type for von Mises functionals and \(U\)-statistics. In: Proceedings of the St. Petersburg mathematical society 7:124–167. English translation in AMS Translations ser.2 203, 2001, 107–146Google Scholar
  23. Nikitin YY, Tchirina AV (1996) Bahadur efficiency and local optimality of a test for the exponential distribution based on the Gini statistic. Stat. Methodol Appl 5(1):163–175zbMATHGoogle Scholar
  24. Nikitin YY, Volkova KY (2010) Asymptotic efficiency of exponentiality tests based on order statistics characterization. Georgian Math J 17(4):749–763MathSciNetzbMATHGoogle Scholar
  25. Obradović M (2015) Three characterizations involving median of sample of size three. J Stat Theory Appl. arXiv:1412.2563v1
  26. Pyke R (1965) Spacings. J R Stat Soc Ser B Stat Methodol 27(3):395–449MathSciNetzbMATHGoogle Scholar
  27. Silverman BW (1983) Convergence of a class of empirical distribution functions of dependent random variables. Ann Probab 11:745–751CrossRefMathSciNetzbMATHGoogle Scholar
  28. Volkova KY (2010) On asymptotic efficiency of exponentiality tests based on Rossbergs characterization. J Math Sci (NY) 167(4):486–494CrossRefMathSciNetzbMATHGoogle Scholar
  29. Wieand HS (1976) A condition under which the Pitman and Bahadur approaches to efficiency coincide. Ann Stat 4:1003–1011CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Faculty of MathematicsUniveristy of BelgradeBelgradeSerbia

Personalised recommendations