Skip to main content
SpringerLink
Log in

New class of exponentiality tests based on U-empirical Laplace transform

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

In this paper, a new class of goodness of fit tests for exponential distribution is proposed. The tests use the equidistribution characterizations of exponential distribution. Based on the U-empirical Laplace transforms of equidistributed statistics, test statistics of the integral type are formed. They are U-statistics with estimated parameters. Their asymptotic properties are derived. Two families of exponentiality tests from this class, based on two selected characterizations, are presented. The approximate Bahadur efficiency is used to assess their quality. Finally, their simulated powers are calculated and the tests are compared with different exponentiality tests.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • 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):681–689

    Article  MathSciNet  MATH  Google Scholar 

  • Ahsanullah M, Hamedani GG (2010) Exponential distribution: theory and methods. NOVA Science, New York

    MATH  Google Scholar 

  • Angus JE (1982) Goodness-of-fit test for exponentiality based on loss of memory type functional equation. J Stat Plan Inference 6(3):241–251

    Article  MathSciNet  MATH  Google Scholar 

  • Arnold BC, Balakrishnan N, Nagaraja HN (2008) A first course in order statistics. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Bahadur RR (1971) Some limit theorems in statistics. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Balakrishnan N, Rao CR (1998) Order statistics, theory and methods. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Baringhaus L, Henze N (1991) A class of consistent tests for exponentiality based on the empirical Laplace transform. Ann Inst Stat Math 43(3):551–564

    Article  MathSciNet  MATH  Google Scholar 

  • Baringhaus L, Henze N (2000) Tests of fit for exponentiality based on a characterization via the mean residual life function. Stat Pap 41(2):225–236

    Article  MathSciNet  MATH  Google Scholar 

  • Cox DR, Oakes D (1984) Analysis of survival data. Chapman and Hall, New York

    Google Scholar 

  • D’Agostino R, Stephens M (1986) Goodness-of-fit techniques. Marcel Dekker Inc., New York

    MATH  Google Scholar 

  • Desu MM (1971) A characterization of the exponential distribution by order statistics. Ann Math Stat 42(2):837–838

    Article  MATH  Google Scholar 

  • Epps TW, Pulley LB (1986) A test for exponentiality vs. monotone hazard alternatives derived from the empirical characteristic function. J R Stat Soc Ser B. Stat Methodol 48(2):206–213

    MathSciNet  MATH  Google Scholar 

  • Galambos J, Kotz S (1978) Characterizations of probability distributions. Springer, Berlin

    Book  MATH  Google Scholar 

  • Gürtler N, Henze N (2000) Goodness-of-fit tests for the Cauchy distribution based on the empirical characteristic function. Ann Inst Stat Math 52(2):267–286

    Article  MathSciNet  MATH  Google Scholar 

  • Henze N (1993) A new flexible class of omnibus tests for exponentiality. Commun Stat Theory Methods 22(1):115–133

    Article  MathSciNet  MATH  Google Scholar 

  • Henze N, Klar B (2002) Goodness-of-fit tests for the inverse Gaussian distribution based on the empirical Laplace transform. Ann Inst Stat Math 54(2):425–444

    Article  MathSciNet  MATH  Google Scholar 

  • Henze N, Meintanis SG (2002) Tests of fit for exponentiality based on the empirical Laplace transform. Statistics 36(2):147–161

    Article  MathSciNet  MATH  Google Scholar 

  • Henze N, Meintanis SG (2002) Goodness-of-fit tests based on a new characterization of the exponential distribution. Commun Stat Theory Methods 31(9):1479–1497

    Article  MathSciNet  MATH  Google Scholar 

  • Henze N, Meintanis SG (2005) Recent and classical tests for exponentiality: a partial review with comparisons. Metrika 61(1):29–45

    Article  MathSciNet  MATH  Google Scholar 

  • Henze N, Meintanis SG, Ebner B (2012) Goodness-of-fit tests for the gamma distribution based on the empirical Laplace transform. Commun Stat Theory Methods 41(9):1543–1556

    Article  MathSciNet  MATH  Google Scholar 

  • Henze N, Nikitin Y, Ebner B (2009) Integral distribution-free statistics of Lp-type and their asymptotic comparison. Comput Stat Data Anal 53(9):3426–3438

    Article  MathSciNet  MATH  Google Scholar 

  • Hoeffding W (1948) A class of statistics with asymptotically normal distribution. Ann Math Stat 19(3):293–395

    Article  MathSciNet  MATH  Google Scholar 

  • Iverson HK, Randles RH (1989) The effects on convergence of substituting parameter estimates into U-statistics and other families of statistics. Prob Theory Relat Fields 81(3):453–471

    Article  MathSciNet  MATH  Google Scholar 

  • Jansen van Rensburg HM, Swanepoel JWH (2008) A class of goodness-of-fit tests based on a new characterization of the exponential distribution. J Nonparametr Stat 20(6):539–551

    Article  MathSciNet  MATH  Google Scholar 

  • Jovanović M, Milošević B, Nikitin YY, Obradović M, Volkova KY (2015) Tests of exponentiality based on Arnold–Villasenor characterization, and their efficiencies. Comput Stat Data Anal 90:100–113

    Article  MathSciNet  Google Scholar 

  • Klar B (2001) Goodness-of-fit tests for the exponential and the normal distribution based on the integrated distribution function. Ann Inst Stat Math 53:338–353

    Article  MathSciNet  MATH  Google Scholar 

  • Koul HL (1977) A test for new better than used. Commun Stat Theory Methods 6(6):563–574

    Article  MathSciNet  MATH  Google Scholar 

  • Koul HL (1978) Testing for new is better than used in expectation. Commun Stat Theory Methods 7(7):685–701

    Article  MathSciNet  MATH  Google Scholar 

  • Meintanis SG, Iliopoulos G (2003) Tests of fit for the Rayleigh distribution based on the empirical Laplace transform. Ann Inst Stat Math 55(1):137–151

    MathSciNet  MATH  Google Scholar 

  • Meintanis SG, Nikitin YY, Tchirina AV (2007) Testing exponentiality against a class of alternatives which includes the RNBUE distributions based on the empirical Laplace transform. J Math Sci (NY) 145(2):4871–4879

    Article  Google Scholar 

  • Milošević B (2016) Asymptotic efficiency of new exponentiality tests based on a characterization. Metrika 79(2):221–236

    Article  MathSciNet  MATH  Google Scholar 

  • Nikitin YY (1995) Asymptotic efficiency of nonparametric tests. Cambridge University Press, New York

    Book  MATH  Google Scholar 

  • Nikitin YY (1996) Bahadur efficiency of test of exponentiality based on loss of memory type functional equation. J Nonparametr Stat 6(1):13–26

    Article  MathSciNet  MATH  Google Scholar 

  • 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):163–175

    MATH  Google Scholar 

  • Nikitin YY, Peaucelle I (2004) Efficiency and local optimality of nonparametric tests based on U- and V-statistics. Metron 62(2):185–200

    MathSciNet  Google Scholar 

  • Nikitin YY, Volkova KY (2010) Asymptotic efficiency of exponentiality tests based on order statistics characterization. Georgian Math J 17(4):749–763

    MathSciNet  MATH  Google Scholar 

  • Puri PS, Rubin H, Characterization A (1970) Based on absolute difference of two I.I.D. random variables. Ann Math Stat 41(6):2113–2122

    Article  MathSciNet  MATH  Google Scholar 

  • Randles RH (1982) On the asymptotic normality of statistics with estimated parameters. Ann Stat 10(2):462–474

    Article  MathSciNet  MATH  Google Scholar 

  • Volkova KY (2010) On asymptotic efficiency of exponentiality tests based on Rossbergs characterization. J Math Sci (NY) 167(4):486–494

    Article  MathSciNet  MATH  Google Scholar 

  • Wieand HS (1976) A condition under which the Pitman and Bahadur approaches to efficiency coincide. Ann Stat 4(5):1003–1011

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the anonymous referees for their important remarks which significantly improved the quality of the paper.

Author information

Authors and Affiliations

  1. Studentski Trg 16, Belgrade, Serbia

    Bojana Milošević & Marko Obradović

Authors
  1. Bojana Milošević
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marko Obradović
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marko Obradović.

Additional information

The research of B. Milošević is supported by Ministry of Education, Science and Technological Development of Republic of Serbia Grant no. 174012.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Milošević, B., Obradović, M. New class of exponentiality tests based on U-empirical Laplace transform. Stat Papers 57, 977–990 (2016). https://doi.org/10.1007/s00362-016-0818-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-016-0818-z

Keywords

Search

Navigation