Abstract
An updated version of the cumulative residual extropy (CRJ) with some dynamic features is introduced in this study as normalized dynamic survival extropy (NDSE). We observe that NDSE grabs attention in the reliability theory also, since NDSE of the relevant random variable in the age replacement model is always equal to the CRJ of that variable. This paper proposes an NDSE based non-parametric test that can be used to see if a data set follows an exponential trend. The proposed test makes itself unique due to its exact distribution of test statistics, scale invariant property, consistency, asymptotic normality, great power and simplicity in calculation. Being aware of the problem of censored observation, we explained how our test will be applied to it. We also provide an extensive power comparison with 5 different tests, each taking different alternatives, and a few real-life examples.
Acknowledgement
The authors are thankful to reviewers for carefully checking the mathematical accuracy of the manuscript.
-
(Communicated by Gejza Wimmer)
References
[1] Abbasnejad, M.—Arghami, N. R.—Tavakoli, M.: A goodness-of-fit test for exponentiality based on Lin-Wong information, J. Iran. Stat. Soc. 11 (2012), 191–202.Search in Google Scholar
[2] Abushal, T.: Estimation of the unknown parameters for the compound rayleigh distribution based on progressive first-failure-censored sampling, Open J. Stat. 1 (2011), 161–171.10.4236/ojs.2011.13020Search in Google Scholar
[3] Baratpour, S.—Rad, A. H.: Testing goodness-of-fit for exponential distribution based on cumulative residual entropy, Comm. Statist. Theory Methods 41 (2012), 1387–1396.10.1080/03610926.2010.542857Search in Google Scholar
[4] Baringhaus, L.—Henze, N.: A class of consistent tests for exponentiality based on the empirical Laplace transform, Ann. Inst. Stat. Math. 43 (1991), 551–664.10.1007/BF00053372Search in Google Scholar
[5] Baringhaus, L.—Henze, N.: Tests of fit for exponentiality based on a characterization via the mean residual life function, Stat. Papers 41 (2000), 225–236.10.1007/BF02926105Search in Google Scholar
[6] Baringhaus, L.—Henze, N.: A new weighted integral goodness-of-fit statistic for exponentiality, Stat. Probab. Lett. 78 (2008), 1006–1016.10.1016/j.spl.2007.09.060Search in Google Scholar
[7] Barlow, R. E.—Proschan, F.: Mathematical Theory of Reliability, SIAM, 1996.10.1137/1.9781611971194Search in Google Scholar
[8] Box, G. E.: Some theorems on quadratic forms applied in the study of analysis of variance problems, I. Effect of inequality of variance in the one-way classification, Ann. Math. Stat. 25 (1954), 290–302.10.1214/aoms/1177728786Search in Google Scholar
[9] Choi, B.—Kim, K.—Song, S. H.: Goodness-of-fit test for exponentiality based on Kullback–Leibler information, Comm. Statist. Simulation Comput. 33 (2004), 525–536.10.1081/SAC-120037250Search in Google Scholar
[10] Datta, S.—Bandyopadhyay, D.—Satten, G. A.: Inverse probability of censoring weighted U-statistics for right-censored data with an application to testing hypotheses, Scand. J. Stat. 37 (2010), 680–700.10.1111/j.1467-9469.2010.00697.xSearch in Google Scholar
[11] Deshpande, V. J.: A class of tests for exponentiality against increasing failure rate average alternatives, Biometrika 70 (1983), 514–518.10.1093/biomet/70.2.514Search in Google Scholar
[12] Dube, S.—Pradhan, B.—Kundu, D.: Parameter estimation of the hybrid censored log-normal distribution, J. Stat. Comput. Simul. 81 (2011), 275–287.10.1080/00949650903292650Search in Google Scholar
[13] Epps, T. W.—Pulley, L. B.: A test of exponentiality vs. monotone hazard alternatives derived from the empirical characteristic function, J. R. Stat. Soc. B 48 (1986), 206–213.10.1111/j.2517-6161.1986.tb01403.xSearch in Google Scholar
[14] Finkelstein, J. M.—Schafer, R. E.: Improved goodness-of-fit tests, Biometrika 58 (1971), 641–645.10.1093/biomet/58.3.641Search in Google Scholar
[15] Gail, M. H.—Gastwirth, J. L.: A scale-free goodness-of-fit test for the exponential distribution based on the Gini statistic, J. R. Stat. Soc. B 40 (1978), 350–357.10.1111/j.2517-6161.1978.tb01048.xSearch in Google Scholar
[16] Gnedenko, B. V.—Belyayev, Y. U. K.—Solovyev, A. D.: Mathematical Models of Reliability Theory, Academic Press, London, 1969.Search in Google Scholar
[17] Harris, C. M.: A note on testing for exponentiality, Nav. Res. Logist. Q. 23 (1976), 169–175.10.1002/nav.3800230116Search in Google Scholar
[18] Henze, N.: A new flexible class of omnibus tests for exponentiality, Comm. Statist. Theory Methods 22 (1993), 115–133.10.1080/03610929308831009Search in Google Scholar
[19] Henze, N.—Klar, B.: Testing exponentiality against the L class of life distributions, Math. Methods Stat. 10 (2001), 232–246.Search in Google Scholar
[20] Jahanshahi, S.—Zarei, H.—Khammar, A.: On cumulative residual extropy, Probab. Engrg. Inform. Sci. 34 (2020), 605–625.10.1017/S0269964819000196Search in Google Scholar
[21] Jaynes, E. T.: Information theory and statistical mechanics, Phys. Rev. 106 (1957), 620–630.10.1103/PhysRev.106.620Search in Google Scholar
[22] Kasilingam, D.—Sathiya Prabhakaran, S. P.—Rajendran, D. K.—Rajagopal, V.—Santhosh Kumar, T.—Soundararaj, A.: Exploring the growth of Covid-19 cases using exponential modelling across 42 countries and predicting signs of early containment using machine learning, Transboundary and Emerging Diseases 68(3) (2021), 1001–1018.10.1111/tbed.13764Search in Google Scholar PubMed PubMed Central
[23] Kattumannil, S. K.—Anisha, P.: A simple non-parametric test for decreasing mean time to failure, Statist. Papers 60 (2019), 73–87.10.1007/s00362-016-0827-ySearch in Google Scholar
[24] Kayid, M.—Ahmad, I. A.—Izadkhah, S.—Abouammoh, A. M.: Further results involving the mean time to failure order, and the decreasing mean time to failure class, IEEE Transactions on Reliability 62 (2013), 670–678.10.1109/TR.2013.2270423Search in Google Scholar
[25] Koul, H. L.—Susarla, V.: Testing for new better than used in expectation with incomplete data, J. Amer. Statist. Assoc. 75 (1980), 952–956.10.1080/01621459.1980.10477578Search in Google Scholar
[26] Koul, H.—Susarla, V.—Van Ryzin, J.: Regression analysis with randomly right-censored data, Annals Statist. 9 (1981), 1276–1288.10.1214/aos/1176345644Search in Google Scholar
[27] Lad, F.—Sanfilippo, G.—Agrò, G.: Extropy: Complementary dual of entropy, Statist. Sci. 30 (2015), 40–58.10.1214/14-STS430Search in Google Scholar
[28] Lawless, J. F.: Statistical Models and Methods for Lifetime Data, John Wiley & Sons, 2011.Search in Google Scholar
[29] Lehmann, E. L.: Consistency and unbiasedness of certain nonparametric tests, The Annals of Mathematical Statistics 22 (1951), 165–179.10.1214/aoms/1177729639Search in Google Scholar
[30] Ossai, E. O.—Madukaife, M. S.—Oladugba, A. V.: A review of tests for exponentiality with Monte Carlo comparisons, J. Appl. Stat. 49(5) (2020), 1277–1304.10.1080/02664763.2020.1854202Search in Google Scholar PubMed PubMed Central
[31] Proschan, F.: Theoretical explanation of observed decreasing failure rate, Technometrics 5 (1963), 375–383.10.1080/00401706.1963.10490105Search in Google Scholar
[32] Rotnitzky, A.—Robins, J.: Inverse probability weighted estimation in survival analysis, Encyclopedia of Biostatistics 4 (2005), 2619–2625.10.1002/0470011815.b2a11040Search in Google Scholar
[33] Sathar, E. A.—Nair, D. R.: On dynamic failure extropy, J. Indian Soc. Probab. Stat. 21 (2020), 287–313.10.1007/s41096-020-00083-xSearch in Google Scholar
[34] Sathar, E. A.—Nair, D. R.: On dynamic survival extropy, Comm. Statist. Theory Methods 50 (2021), 1295–1313.10.1080/03610926.2019.1649426Search in Google Scholar
[35] Shannon, C. E.: A mathematical theory of communications, Bell Syst. Tech. J. 27 (1948), 379–423.10.1002/j.1538-7305.1948.tb01338.xSearch in Google Scholar
[36] Soest, J. V.: Some goodness of fit tests for exponential distributions, Stat. Neerl. 23 (1969), 41–51.10.1111/j.1467-9574.1969.tb00072.xSearch in Google Scholar
[37] Zardasht, V.—Parsi, S.—Mousazadeh, M.: On empirical cumulative residual entropy and a goodness-of-fit test for exponentiality, Statistical Papers 56 (2015), 677–688.10.1007/s00362-014-0603-9Search in Google Scholar
[38] Zhao, H.—Tsiatis, A. A.: Estimating mean quality adjusted lifetime with censored data, Sankhya B 62 (2000), 175–188.Search in Google Scholar
© 2025 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Joins of normal matrices, their spectrum, and applications
- On isbell’s density theorem for bitopological pointfree spaces II
- Every positive integer is a sum of at most n + 2 centered n-gonal numbers
- A generalisation of q-additive functions
- Generalised class groups in dihedral and fake ℤp-extensions
- Singular value bounds with applications to norm and numerical radius inequalities
- Coefficient bounds for convex functions associated with cosine function
- On solutions to q-difference equations for q-appell functions in the spirit of Olsson and Exton
- Numerical implementation of solving a boundary value problem including both delay and parameter
- Solutions of second order iterative boundary value problems with nonhomogeneous boundary conditions
- Global attractivity of nonlinear delay dynamic equations on time scales via Lyapunov functional method
- On pseudo almost periodic solutions of the parabolic-elliptic Keller-Segel systems
- A note on derivations into annihilators of the ideals of banach algebras
- Characterization of nonlinear mixed skew lie and jordan n-Type derivation on ∗-Algebras
- Linear and uniformly continuous surjections between Cp-spaces over metrizable spaces
- A goodness-of-fit test for testing exponentiality based on normalized dynamic survival extropy
- Monotonicity results of ratio between two normalized remainders of Maclaurin series expansion for square of tangent function
Articles in the same Issue
- Joins of normal matrices, their spectrum, and applications
- On isbell’s density theorem for bitopological pointfree spaces II
- Every positive integer is a sum of at most n + 2 centered n-gonal numbers
- A generalisation of q-additive functions
- Generalised class groups in dihedral and fake ℤp-extensions
- Singular value bounds with applications to norm and numerical radius inequalities
- Coefficient bounds for convex functions associated with cosine function
- On solutions to q-difference equations for q-appell functions in the spirit of Olsson and Exton
- Numerical implementation of solving a boundary value problem including both delay and parameter
- Solutions of second order iterative boundary value problems with nonhomogeneous boundary conditions
- Global attractivity of nonlinear delay dynamic equations on time scales via Lyapunov functional method
- On pseudo almost periodic solutions of the parabolic-elliptic Keller-Segel systems
- A note on derivations into annihilators of the ideals of banach algebras
- Characterization of nonlinear mixed skew lie and jordan n-Type derivation on ∗-Algebras
- Linear and uniformly continuous surjections between Cp-spaces over metrizable spaces
- A goodness-of-fit test for testing exponentiality based on normalized dynamic survival extropy
- Monotonicity results of ratio between two normalized remainders of Maclaurin series expansion for square of tangent function