# Article

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Keywords:
goodness-of-fit; asymptotic efficiency; V-statistics; characterization; test for exponentiality
Summary:
We present new goodness-of-fit tests for the exponential distribution based on equidistribution type characterizations. For the construction of the test statistics, we employ an \$L^2\$-distance between the corresponding V-empirical distribution functions. The resulting test statistics are V-statistics, free of the scale parameter. \endgraf The quality of the tests is assessed through local Bahadur efficiencies as well as the empirical power for small and moderate sample sizes. According to both criteria, for many common alternatives, our tests perform better than the integral and Kolmogorov-type tests based on the same characterizations.
References:
[1] Ahsanullah, M.: Characterizations of Univariate Continuous Distributions. Atlantis Studies in Probability and Statistics 7. Atlantis Press, Amsterdam (2017). DOI 10.2991/978-94-6239-139-0 | MR 3618798 | Zbl 1366.62005
[2] Bahadur, R. R.: Some Limit Theorems in Statistics. CBMS-NSF Regional Conference Series in Applied Mathematics 4, SIAM, Philadelphia (1971). DOI 10.1137/1.9781611970630 | MR 0315820 | Zbl 0257.62015
[3] Bönner, N., Kirschner, H.-P.: Note on conditions for weak convergence of von Mises' differentiable statistical functions. Ann. Stat. 5 (1977), 405-407. DOI 10.1214/aos/1176343807 | MR 0428545 | Zbl 0358.62018
[4] Božin, V., Milošević, B., Nikitin, Ya. Yu., Obradović, M.: New characterization-based symmetry tests. Bull. Malays. Math. Sci. Soc. (2) 43 (2020), 297-320. DOI 10.1007/s40840-018-0680-3 | MR 4045140 | Zbl 07173767
[5] Cuparić, M., Milošević, B., Obradović, M.: New consistent exponentiality tests based on \$V\$-empirical Laplace transforms with comparison of efficiencies. Available at https://arxiv.org/abs/1904.00840 (2019). MR 3973667
[6] Cuparić, M., Milošević, B., Obradović, M.: New \$L^2\$-type exponentiality tests. SORT 43 (2019), 25-50. DOI 10.2436/20.8080.02.78 | MR 3973667 | Zbl 1420.62192
[7] Desu, M. M.: A characterization of the exponential distribution by order statistics. Ann. Math. Stat. 42 (1971), 837-838. DOI 10.1214/aoms/1177693444 | Zbl 0216.47502
[8] Galambos, J., Kotz, S.: Characterizations of Probability Distributions: A Unified Approach with an Emphasis on Exponential and Related Models. Lecture Notes in Mathematics 675, Springer, Berlin (1978). DOI 10.1007/BFb0069530 | MR 0513423 | Zbl 0381.62011
[9] Jovanović, M., Milošević, B., Nikitin, Ya. Yu., Obradović, M., Volkova, K. Yu.: Tests of exponentiality based on Arnold-Villasenor characterization and their efficiencies. Comput. Stat. Data Anal. 90 (2015), 100-113. DOI 10.1016/j.csda.2015.03.019 | MR 3354832 | Zbl 06921408
[10] Kagan, A. M., Linnik, Yu. V., Rao, C. R.: Characterization Problems in Mathematical Statistics. Wiley Series in Probability and Mathematical Statistics, Wiley, New York (1973). MR 0346969 | Zbl 0271.62002
[11] Kato, T.: Perturbation Theory of Linear Operators. Grundlehren der mathematischen Wissenschaften 132, Springer, Berlin (1980). DOI 10.1007/978-3-662-12678-3 | MR 1335452 | Zbl 0435.47001
[12] Korolyuk, V. S., Borovskikh, Yu. V.: Theory of \$U\$-Statistics. Mathematics and Its Applications (Dordrecht) 273, Kluwer Academic, Dordrecht (1994). DOI 10.1007/978-94-017-3515-5 | MR 1472486 | Zbl 0785.60015
[13] Milošević, B.: Asymptotic efficiency of new exponentiality tests based on a characterization. Metrika 79 (2016), 221-236. DOI 10.1007/s00184-015-0552-x | MR 3451377 | Zbl 1341.60020
[14] Milošević, B., Obradović, M.: New class of exponentiality tests based on U-empirical Laplace transform. Stat. Pap. 57 (2016), 977-990. DOI 10.1007/s00362-016-0818-z | MR 3571184 | Zbl 1416.62251
[15] Milošević, B., Obradović, M.: Some characterization based exponentiality tests and their Bahadur efficiencies. Publ. Inst. Math., Nouv. Sér. 100 (2016), 107-117. DOI 10.2298/PIM1614107M | MR 3586684 | Zbl 06749641
[16] Milošević, B., Obradović, M.: Some characterizations of the exponential distribution based on order statistics. Appl. Anal. Discrete Math. 10 (2016), 394-407. DOI 10.2298/AADM160421010M | MR 3586748 | Zbl 06750210
[17] Nikitin, Ya. Yu.: Asymptotic Efficiency of Nonparametric Tests. Cambridge University Press, Cambridge (1995). DOI 10.1017/CBO9780511530081 | MR 1335235 | Zbl 0879.62045
[18] Nikitin, Ya. Yu., Peaucelle, I.: Efficiency and local optimality of nonparametric tests based on U- and V-statistics. Metron 62 (2004), 185-200. MR 2102099 | Zbl 1416.62253
[19] Nikitin, Ya. Yu., Volkova, K. Yu.: Asymptotic efficiency of exponentiality tests based on order statistics characterization. Georgian Math. J. 17 (2010), 749-763. DOI 10.1515/gmj.2010.034 | MR 2746191 | Zbl 1205.62052
[20] Nikitin, Ya. Yu., Volkova, K. Yu.: Efficiency of exponentiality tests based on a special property of exponential distribution. Math. Methods Stat. 25 (2016), 54-66. DOI 10.3103/S1066530716010038 | MR 3480610 | Zbl 1345.60019
[21] Polyanin, A. D., Manzhirov, A. V.: Handbook of Integral Equations. CRC Press, Boca Raton (2008). DOI 10.1201/9781420010558 | MR 2404728 | Zbl 1154.45001
[22] Puri, P. S., Rubin, H.: A characterization based on the absolute difference of two i.i.d. random variables. Ann. Math. Stat. 41 (1970), 2113-2122. DOI 10.1214/aoms/1177696709 | MR 0293761 | Zbl 0225.60007

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