276
Views
2
CrossRef citations to date
0
Altmetric
Review Article

A review of tests for exponentiality with Monte Carlo comparisons

, ORCID Icon &
Pages 1277-1304 | Received 26 Jan 2020, Accepted 14 Nov 2020, Published online: 04 Dec 2020

References

  • M. Abbasnejad, N.R. Arghami, and M. Tavakoli, A goodness-of-fit test for exponentiality based on Lin–Wong information. J. Iran. Stat. Soc. 11 (2012), pp. 191–202.
  • A.A. Abdel-Aziz, On testing exponentiality against RNBRUE alternatives. Appl. Math. Sci. 1(35) (2007), pp. 1725–1736.
  • I. B. Abdul-Moniem, Testing exponentiality versus NBUASI based on moments inequalities. InterStat 3 (2008).
  • I.A. Ahmad, A simple and more efficient new approach to life testing. Commun. Stat. Theory Methods 33(9) (2004), pp. 2199–2215.
  • I.A. Ahmad, A. Ahmed, I. Elbatal, and M. Kayid, An aging notion derived from the increasing convex ordering: the NBUCA class. J. Stat. Plan. Inference. 136 (2006), pp. 555–569.
  • V. Ahrari, A.H. Rad, and S. Baratpour, Exponentiality test based on alpha-divergence and gamma-divergence. Commun. Stat.-Simul. Comput. 48(4) (2019), pp. 1138–1152.
  • M. Ahsanullah, On a characterization of the exponential distribution by spacings. Ann. Inst. Stat. Math. Part A 30 (1978), pp. 163–166.
  • B. Al – Zahrani, and J. Stoyanov, Moment inequalities for HNBRUE with hypothesis testing application. Stat. Methodol. 7 (2010), pp. 58–67.
  • F.H. Al-Gashgari, A.I. Shawky, and M.A.W. Mahmoud, A nonparametric test for testing exponentiality against NBUCA class of life distributions based on Laplace transform. Qual. Reliab. Eng. Int. 32(1) (2016), pp. 29–36.
  • J.S. Allison, L. Santana, N. Smit, and I.J.H. Visagie, An ‘apples to apples’ comparison of various tests for exponentiality. Comput. Stat. 32(4) (2017), pp. 1241–1283.
  • E.-E.A.A. Aly, On testing exponentiality against IFRA alternatives. Metrika 36 (1989), pp. 255–267.
  • E.-E.A.A. Aly, Tests for monotonicity properties of the mean residual life function. Scand. J. Stat. 17 (1990), pp. 189–200.
  • S. Amari, Differential-Geometrical Methods in Statistics, Springer Verlag, Berlin, 1985.
  • S. Amari, Alpha-divergence is unique, belonging to both f-divergence and Bregman divergence classes. IEEE Trans. Inf. Theory 55 (2009), pp. 4925–4931.
  • S. Amari, and H. Nagaoka, Methods of Information Geometry, Oxford University Press, New York, 2000.
  • M.Z. Anis, On testing exponentiality against DMRL alternatives. Econ. Quality Control 25 (2010), pp. 281–299.
  • M.Z. Anis, and M. Mitra, A simple test of exponentiality against NWBUE family of life distributions. Appl. Stoch. Models. Bus. Ind. 21 (2005), pp. 45–53.
  • M.Z. Anis, and M. Mitra, A generalized Hollander-Proschan type test for NBUE alternatives. Statist. Probab. Lett. 81(1) (2011), pp. 126–132.
  • S. Ascher, A survey of tests for exponentiality. Commun. Stat.-Theory Methods 19 (1990), pp. 1811–1825.
  • G. Asha, and N.U. Nair, Reliability properties of mean time to failure in age replacement models. Int. J. Reliab. Qual. Saf. Eng. 17 (2010), pp. 15–26.
  • M.A. Atallah, M.A.W. Mahmoud, and B.M. Al- Zahrani, A new test for exponentiality versus NBUmgf life distributions based on Laplace transform. Qual. Reliab. Eng. Int. 30(8) (2014), pp. 1353–1359.
  • S. Baratpour, and A. Habibirad, Testing goodness – of – fit for exponential distribution based on cumulative residual entropy. Commun. Stat.-Theory Methods 41 (2012), pp. 1387–1396.
  • L. Baringhaus, and N. Henze, A class of consistent tests for exponentiality based on the empirical Laplace transform. Ann. Inst. Stat. Math. 43 (1991), pp. 551–664.
  • L. Baringhaus, and N. Henze, Tests of fit for exponentiality based on a characterization via the mean residual life function. Stat. Papers 41 (2000), pp. 225–236.
  • L. Baringhaus, and N. Henze, A new weighted integral goodness-of-fit statistic for exponentiality. Stat. Probab. Lett. 78 (2008), pp. 1006–1016.
  • L. Baringhaus, and F. Taherizadeh, Empirical Hankel transforms and its applications to goodness-of-fit tests. J. Multivar. Anal. 101 (2010), pp. 1445–1457.
  • L. Baringhaus, and F. Taherizadeh, A K-S type test for exponentiality based on empirical Hankel transforms. Commun. Stat.-Theory Methods 42(20) (2013), pp. 3781–3792.
  • R.E. Barlow, and F. Proschan, Mathematical Theory of Reliability, Wiley, New York, 1965.
  • S. Basu, and M. Mitra, Testing exponentiality against Laplace order dominance. Statistics (Ber) 36 (2002), pp. 223–229.
  • F. Belzunce, J. Candel, and J.M. Ruiz, Testing the stochastic order and the IFR, DFR, NBU, NWU ageing classes. IEEE Trans. Reliab. 47 (1998), pp. 285–296.
  • F. Belzunce, J. Candel, and J.M. Ruiz, Testing mean residual alternatives by dispersion of residual lives. J. Stat. Plan. Inference. 86 (2000), pp. 113–127.
  • B. Bergman, and B. Klefsjö, A family of test statistics for detecting monotone mean residual life. J. Stat. Plan. Inference. 21 (1989), pp. 161–178.
  • D. Bhattacharyya, R.A. Khan, and M. Mitra, A test of exponentiality against DMTTF alternatives via L-statistics. Stat. Probab. Lett. (2020). doi:https://doi.org/10.1016/j.spl.2020.108853.
  • G. Chaudhuri, Testing exponentiality against L-distributions. J. Stat. Plan. Inference. 64 (1997), pp. 249–255.
  • H. Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on a sum of observations. Ann. Math. Stat. 23 (1952), pp. 493–507.
  • R. D’Agostino, and M. Stephens, Goodness-of-fit Techniques, Marcel Dekker, New York, 1986.
  • V.J. Deshpande, A class of tests for exponentiality against increasing failure rate average alternatives. Biometrika 70 (1983), pp. 514–518.
  • L.S. Diab, and E.S. El-Atfy, A moment inequality for overall decreasing life class of life distributions with hypotheses testing applications. Int. J. Math. Stat. Invent. 5(1) (2017), pp. 62–71.
  • A.H. El-Bassiouny, On testing exponentiality against IFRA alternatives. Appl. Math. Comput. 146 (2003), pp. 445–453.
  • I. Elbatal, On testing statistics of renewal new better than renewal used class of life distributions. Int. J. Contemp. Math. Sci. 4(1) (2009), pp. 17–29.
  • T.W. Epps, and L.B. Pulley, A test of exponentiality vs. monotone – hazard alternatives derived from the empirical characteristic function. J. R. Stat. Soc. B 48 (1986), pp. 206–213.
  • B. Epstein, Tests for the validity of the assumption that the underlying distribution of life is exponential (part I). Technometrics. 2(1) (1960a), pp. 83–107.
  • B. Epstein, Tests for the validity of the assumption that the underlying distribution of life is exponential (part II). Technometrics. 2(2) (1960b), pp. 167–183.
  • J.M. Fernandez – Ponce, and M.R. Rodriguez – Grinolo, Testing exponentiality against NBUE distributions with an application in environmental extremes. Stoch. Environ. Res. Risk. Assess. 29 (2015), pp. 679–692.
  • B. V. Frozini, On the distribution and power of a goodness-of-fit statistic with parametric and nonparametric applications. Goodness-of-fit, (Ed. by Revesz P., Sarkadi K., Sen P.K.) North-Holland, New York. (1987).
  • M.H. Gail, and J.L. Gastwirth, A scale – free goodness – of – fit test for the exponential distribution based on the Gini statistic. J. R. Stat. Soc. B 40 (1978a), pp. 350–357.
  • M.H. Gail, and J.L. Gastwirth, A scale-free goodness-of-fit test for the exponential distribution based on the Lorenz curve. J. Am. Stat. Assoc. 73 (1978b), pp. 787–793.
  • S. Ghosh, and M. Mitra, A Hollander-Proschan type test when ageing is not monotone. Stat. Probab. Lett. 121 (2017a), pp. 119–127.
  • S. Ghosh, and M. Mitra, A weighted integral approach to testing against HNBUE alternatives. Stat. Probab. Lett. 129 (2017b), pp. 58–64.
  • S. Ghosh, and M. Mitra, A new test for exponentiality against HNBUE alternatives. Commun. Stat. - Theory Methods (2020), doi:https://doi.org/10.1080/03610926.2018.1528370.
  • B.V. Gnedenko, Y.U.K. Belyayev, and A.D. Solovyev, Mathematical Models of Reliability Theory, Academic Press, London, 1969.
  • F. Guess, M. Hollander, and F. Proschan, Testing exponentiality versus a trend change in mean residual life. Ann. Stat. 14(4) (1986), pp. 1388–1398.
  • G.J. Hahn, and S.S. Shapiro, Statistical Models in Engineering, John Wiley and Sons Inc, New York, 1967.
  • C.M. Harris, A note on testing for exponentiality. Nav. Res. Logist. Q. 23 (1976), pp. 169–175.
  • D.L. Hawkins, and S. Kochar, Inference about the transition-point in NBUE-NWUE or NWUE-NBUE models. Sankhyā A 59(1) (1997), pp. 117–132.
  • D.L. Hawkins, S. Kochar, and C. Loader, Testing exponentiality against IDMRL distributions with unknown change point. Ann. Stat. 20 (1992), pp. 280–290.
  • Y.A.S. Hegazy, and J.R. Green, Some new goodness-of-fit tests using order statistics. J. Appl. Stat. 24(3) (1975), pp. 299–308.
  • N. Henze, A new flexible class of omnibus tests for exponentiality. Commun. Stat. – Theory Methods 22 (1993), pp. 115–133.
  • N. Henze, and B. Klar, Testing exponentiality against the L class of life distributions. Math. Methods Stat. 10 (2001), pp. 232–246.
  • N. Henze, and S.G. Meintanis, Tests of fit for exponentiality based on the empirical Laplace transform. Statistics. (Ber) 36 (2002a), pp. 147–161.
  • N. Henze, and S.G. Meintanis, Goodness – of – fit tests based on a new characterization of the exponential distribution. Commun. Stat. – Theory Methods 31 (2002b), pp. 1479–1497.
  • N. Henze, and S.G. Meintanis, Recent and classical tests for exponentiality: a partial review with comparisons. Metrika 61 (2005), pp. 29–45.
  • M. Hollander, and F. Proschan, Tests for the mean residual life. Biometrika 62(3) (1975), pp. 585–593.
  • M. Izadi, and S.F. Manesh, Testing exponentiality against a trend change in mean time to failure in age replacement. Commun. Stat. - Theory Methods (2019), doi:https://doi.org/10.1080/03610926.2019.1702693.
  • S.R. Jammalamadaka, and E. Taufer, Testing exponentiality by comparing the empirical distribution function of the normalized spacings with that of the original data. Nonparametric Stat. 15 (2003), pp. 719–729.
  • S.R. Jammalamadaka, R.C. Tiwari, and J.N. Zalkikar, Testing for exponentiality against IFRA alternatives using a U-statistic process. Braz. J. Probab. Stat. 4(2) (1990), pp. 147–159.
  • S.K. Kattumannil, and P. Anisha, A simple non-parametric test for decreasing mean time to failure. Stat. Papers 60 (2019), pp. 73–87.
  • S.K. Kattumannil, and D.C. Mathew, A Gini-based exact test for exponentiality against NBUE alternatives with censored observations. J. Nonparametr. Stat. 27(4) (2015), pp. 503–515.
  • M. Kayid, I. Ahmad, S. Izadkhah, and A. Abouammoh, Further results involving the mean time to failure order, and the decreasing mean time to failure class. IEEE Trans. Reliab. 62 (2013), pp. 670–678.
  • A.C. Kimber, Tests for the exponential, Weibull and Gumbel distributions based on the stabilized probability plot. Biometrika 72 (1985), pp. 661–663.
  • B. Klar, A class of tests for exponentiality against HNBUE alternatives. Stat. Probab. Lett. 47 (2000), pp. 199–207.
  • B. Klar, Goodness-of-fit tests for the exponential and the normal distribution based on the integrated distribution function. Ann. Inst. Stat. Math. 53(2) (2001), pp. 338–353.
  • B. Klefsjo, Some tests against aging based on the total time on test transform. Commun. Stat.-Theory Methods 12 (1983a), pp. 907–927.
  • B. Klefsjö, Testing exponentiality against HNBUE. Scand. J. Stat. 43 (1983b), pp. 1136–1146.
  • B. Klefsjö, A useful ageing property based on the Laplace transform. J. Appl. Probab. 20 (1983c), pp. 615–626.
  • B. Klefsjo, Testing against a change in the NBUE property. Microelectron. Reliab. 29(4) (1989), pp. 559–570.
  • S.C. Kochar, Testing exponenttality against monotone failure rate average. Commun. Stat. - Theory Methods 14(2) (1985), pp. 381–392.
  • H.L. Koul, A class of tests for testing new better than used. Canad. J. Stat. 6 (1978), pp. 249–271.
  • X. Li, and M. Xu, Reversed hazard rate order of equilibrium distributions and a related aging notion. Stat. Papers 49 (2008), pp. 749–767.
  • F. Louzada, P.L. Ramos, and P.H. Ferreira, Exponential-Poisson distribution: estimation and applications to rainfall and aircraft data with zero occurrence. Commun. Stat. - Simul. Comput. 49(4) (2020), pp. 1024–1043.
  • M.S. Madukaife, An adaptive test for exponentiality based on empirical quantile function. Int. J. Stat. Appl. 9(4) (2019), pp. 111–116.
  • M.A.W. Mahmoud, R.M. El-Sagheer, and W.B.H. Etman, Moments inequalities for NBRUL distributions with hypotheses testing applications. Austr. J. Stat. 47 (2018), pp. 95–104.
  • P. Majumder, and M. Mitra, A test for detecting Laplace order dominance and related Bahadur efficiency issues. Stat. Papers 60 (2019a), pp. 1921–1937.
  • P. Majumder, and M. Mitra, Detecting trend change in hazard functions—an L-statistic approach. Statistical Papers, (2019b) doi:https://doi.org/10.1007/s00362-018-01074-8.
  • J.R. Michael, The stabilized probability plot. Biometrika 70 (1983), pp. 11–17.
  • N. Mimoto, and R. Zitikis, The Atkinson index, the Moran statistic, and testing exponentiality. J. Jpn. Stat. Soc. 38 (2008), pp. 187–205.
  • M. Mitra, and M.Z. Anis, An L – statistic approach to a test of exponentiality against IFR alternatives. J. Stat. Plan. Inference. 138(10) (2008), pp. 3144–3148.
  • M. Mitra, and S.K. Basu, On a nonparametric family of life distributions and its dual. J. Stat. Plan. Inference. 39 (1994), pp. 385–397.
  • P.A.P. Moran, The random division of an interval – Part II. J. R. Stat. Soc. Ser. B 13 (1951), pp. 147–150.
  • M.H. Na, J. Jeon, and D.H. Park, Testing whether failure rate changes its trend with unknown change point. J. Stat. Plan. Inference. 129 (2005), pp. 317–325.
  • M.H. Na, and S. Lee, A family of IDMRL tests with unknown turning point. Statistics (Ber) 37(5) (2003), pp. 457–462.
  • D.H. Park, Testing whether failure rate changes its trend. IEEE Trans. Reliab. 37(4) (1988), pp. 375–378.
  • S. Park, D. Choi, and S. Jung, Kullback-Leibler information of the equilibrium distribution function and its application to goodness of fit test. Commun. Stat. Appl. Methods. 21(2) (2014), pp. 125–134.
  • S. Park, M. Rao, and D.W. Shin, On cumulative residual Kullback-Leibler information. Stat. Probab. Lett. 82 (2012), pp. 2025–2032.
  • G. Patwardhan, Tests for exponentiality. Commun. Stat. Theory Methods 17 (1988), pp. 3705–3722.
  • P. Prajitha, and S. Nadarajah, A Nonparametric test for exponentiality in LIFRA class of life distributions. Sankhya B 76 (2014), pp. 260–275.
  • M. Rahman, and H. Wu, Tests for exponentiality: a comparative study. Am. J. Appl. Math. Stat. 5(4) (2017), pp. 125–135.
  • A. Renyi, On measures of entropy and information. Proceedings of Fourth Berkeley Symposium on Mathematical Statistics and Probability 1 (1961), pp. 547–561.
  • A. P. Rogozhnikov, and B. Y. Lemeshko, (2012) A review of tests for exponentiality. Proceedings of the 11th International Conference of the APEIE, 159–166.
  • M. Sadeghpour, S. Baratpour, and A. Habibirad, Exponentiality test based on Renyi distance between equilibrium distributions. Commun. Stat. - Simul. Comput. 47(10) (2018), pp. 2960–2967.
  • P.G. Sankaran, and N.N. Midhu, Testing exponentiality using mean residual quantile function. Stat. Papers 57 (2016), pp. 235–247.
  • S. Shapiro, and M. Wilk, An analysis of variance test for the exponential distribution (complete samples). Technometrics. 14 (1972), pp. 355–370.
  • N. Sreelakshmi, K. Sudheesh, S.K. Kattumannil, and G. Asha, Quantile based tests for exponentiality against DMRQ and NBUE alternatives. J. Korean. Stat. Soc. 47(2) (2018), pp. 185–200.
  • A.V. Tchirina, Bahadur efficiency and local optimality of a test for exponentiality based on the Moran statistics. J. Math. Sci. 127 (2005), pp. 1812–1819.
  • H. Torabi, N.H. Montazeri, and A. Grané, A wide review on exponentiality tests and two competitive proposals with application on reliability. J. Stat. Comput. Simul. 88(1) (2018), pp. 108–139.
  • K.Y. Volkova, and Y.Y. Nikitin, Exponentiality tests based on Ahsanullah’s characterization and their efficiency. J. Math. Sci. 204(1) (2015), pp. 42–54.
  • P.G. Wong, and S.P. Wong, An extremal quotient test for exponential distributions. Metrika 26(1) (1979), pp. 1–4. www.kaggle.com/datasets, accessed on 16th April 2020.
  • J. Zhang, Powerful goodness-of-fit tests based on the likelihood ratio. J. R. Stat. Soc. Series B 64(2) (2002), pp. 281–294.
  • J. Lin, and S.K.M. Wong, A new directed divergence measure and its characterization. Int. J. General Systems 17 (1990), pp. 73–81.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.