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Keywords:
generalized cumulative entropy; lower record values; reversed relevation transform; stochastic orders; parallel system
generalized cumulative entropy; lower record values; reversed relevation transform; stochastic orders; parallel system
Summary:
Recently, a new concept of entropy called generalized cumulative entropy of order $n$ was introduced and studied in the literature. It is related to the lower record values of a sequence of independent and identically distributed random variables and with the concept of reversed relevation transform. In this paper, we provide some further results for the generalized cumulative entropy such as stochastic orders, bounds and characterization results. Moreover, some characterization results are derived for the dynamic generalized cumulative entropy. Finally, it is shown that the empirical generalized cumulative entropy of an exponential distribution converges to normal distribution.
References:
[1] Arnold, B. C., Balakrishnan, N., Nagaraja, H. N.: A First Course in Order Statistics. Wiley and Sons, New York 1992. DOI 10.1137/1.9780898719062 | MR 1178934
[2] Asadi, M.: A new measure of association between random variables. DOI 10.1007/s00184-017-0620-5
[3] Bagai, L., Kochar, S. C.: On tail ordering and comparison of failure rates. Comm. Stat. Theor. Meth. 15 (1986), 1377-1388. DOI 10.1080/03610928608829189 | MR 0836602
[4] Baratpour, S.: Characterizations based on cumulative residual entropy of first-order statistics. Comm. Stat. Theor. Meth. 39 (2010), 3645-3651. DOI 10.1080/03610920903324841 | MR 2747631
[5] Barlow, R. E., Proschan, F.: Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York 1975. MR 0438625
[6] Bartoszewicz, J.: On a representation of weighted distributions. Stat. Prob. Lett. 79 (2009), 1690-1694. DOI 10.1016/j.spl.2009.04.007 | MR 2547939
[7] Baxter, L. A.: Reliability applications of the relevation transform. Nav. Res. Logist. Q. 29 (1982), 323-330. DOI 10.1002/nav.3800290212 | MR 0681055
[8] Block, H. W., Savits, T. H., Singh, H.: The reversed hazard rate function. Probab. Engrg. Inform. Sci. 12 (1998), 69-90. DOI 10.1017/s0269964800005064 | MR 1492141
[9] Burkschat, M., Navarro, J.: Asymptotic behavior of the hazard rate in systems based on sequential order statistics. Metrika 77 (2014), 965-994. DOI 10.1007/s00184-013-0481-5 | MR 3268630
[10] Chandler, K. N.: The distribution and frequency of record values. J. Royal Stat. Soc. B 14 (1952), 220-228. MR 0053463
[11] Chandra, N. K., Roy, D.: Some results on reversed hazard rate. Probab. Engrg. Inform. Sci. 15 (2001), 95-102. DOI 10.1017/s0269964801151077 | MR 1825537
[12] Crescenzo, A. Di: A probabilistic analogue of the mean value theorem and its applications to reliability theory. J. Appl. Prob. 36 (1999), 706-719. DOI 10.1239/jap/1029349973 | MR 1737047
[13] Crescenzo, A. Di, Longobardi, M.: Entropy-based measure of uncertainty in past lifetime distributions. J. Appl. Prob. 39 (2002), 434-440. DOI 10.1239/jap/1025131441 | MR 1908960
[14] Crescenzo, A. Di, Longobardi, M.: On cumulative entropies. J. Stat. Plan. Infer. 139 (2009), 4072-4087. DOI 10.1016/j.jspi.2009.05.038 | MR 2558351
[15] Crescenzo, A. Di, Martinucci, B., Mulero, J.: A quantile-based probabilistic mean value theorem. Probab. Engrg. Inform. Sci. 30 (2016), 261-280. DOI 10.1017/s0269964815000376 | MR 3478845
[16] Crescenzo, A. Di, Toomaj, A.: Extensions of the past lifetime and its connections to the cumulative entropy. J. Appl. Prob. 52 (2015), 1156-1174. DOI 10.1239/jap/1450802759 | MR 3439178
[17] Hwang, J. S., Lin, G. D.: On a generalized moment problem II. Proc. Amer. Math. Soc. 91 (1984), 577-580. DOI 10.1090/s0002-9939-1984-0746093-4 | MR 0746093
[18] Kamps, U.: Characterizations of distributions by recurrence relations and identities for moments of order statistics. In: Order Statistics: Theory and Methods. Handbook of Statistics 16 (N. Balakrishnan and C. R. Rao, eds.), Elsevier, Amsterdam 1998, pp. 291-311. DOI 10.1016/s0169-7161(98)16012-1 | MR 1668749
[19] Kapodistria, S., Psarrakos, G.: Some extensions of the residual lifetime and its connection to the cumulative residual entropy. Probab. Engrg. Inform. Sci. 26 (2012), 129-146. DOI 10.1017/s0269964811000271 | MR 2880265
[20] Kayal, S.: On generalized cumulative entropies. Probab. Engrg. Inform. Sci. 30 (2016), 640-662. DOI 10.1017/s0269964816000218 | MR 3569139
[21] Kayal, S.: On weighted generalized cumulative residual entropy of order $n$. Meth. Comp. Appl. Prob., online first (2017). DOI 10.1007/s11009-017-9569-0
[22] Klein, I., Mangold, B., Doll, M.: Cumulative paired $\phi$-entropy. Entropy 18 (2016), 248. DOI 10.3390/e18070248 | MR 3550258
[23] Krakowski, M.: The relevation transform and a generalization of the Gamma distribution function. Rev. Française Automat. Informat. Recherche Opérationnelle 7 (1973), 107-120. DOI 10.1051/ro/197307v201071 | MR 0329175
[24] Li, Y., Yu, L., Hu, T.: Probability inequalities for weighted distributions. J. Stat. Plan. Infer. 142 (2012), 1272-1278. DOI 10.1016/j.jspi.2011.11.023 | MR 2879771
[25] Muliere, P., Parmigiani, G., Polson, N.: A note on the residual entropy function. Probab. Engrg. Inform. Sci. 7 (1993), 413-420. DOI 10.1017/s0269964800003016
[26] Navarro, J., Aguila, Y. del, Asadi, M.: Some new results on the cumulative residual entropy. J. Stat. Plan. Infer. 140 (2010), 310-322. DOI 10.1016/j.jspi.2009.07.015 | MR 2568141
[27] Navarro, J., Psarrakos, G.: Characterizations based on generalized cumulative residual entropy functions. Comm. Stat. Theor. Meth. 46 (2017), 1247-1260. DOI 10.1080/03610926.2015.1014111 | MR 3565622
[28] Navarro, J., Rubio, R.: Comparisons of coherent systems with non-identically distributed components. J. Stat. Plann. Infer. 142 (2012), 1310-1319. DOI 10.1016/j.jspi.2011.12.008 | MR 2891483
[29] Navarro, J., Sunoj, S. M., Linu, M. N.: Characterizations of bivariate models using dynamic Kullback-Leibler discrimination measures. Stat. Prob. Lett. 81 (2011), 1594-1598. DOI 10.1016/j.spl.2011.05.016 | MR 2832917
[30] Psarrakos, G., Navarro, J.: Generalized cumulative residual entropy and record values. Metrika 76 (2013), 623-640. DOI 10.1007/s00184-012-0408-6 | MR 3078811
[31] Psarrakos, G., Toomaj, A.: On the generalized cumulative residual entropy with applications in actuarial science. J. Comput. Appl. Math. 309 (2017), 186-199. DOI 10.1016/j.cam.2016.06.037 | MR 3539777
[32] Rao, M.: More on a new concept of entropy and information. J. Theoret. Probab. 18 (2005), 967-981. DOI 10.1007/s10959-005-7541-3 | MR 2289942
[33] Rao, M., Chen, Y., Vemuri, B., Fei, W.: Cumulative residual entropy: a new measure of information. IEEE Trans. Inform. Theory 50 (2004), 1220-1228. DOI 10.1109/tit.2004.828057 | MR 2094878
[34] Rezaei, M., Gholizadeh, B., Izadkhah, S.: On relative reversed hazard rate order. Comm. Stat. Theor. Meth. 44 (2015), 300-308. DOI 10.1080/03610926.2012.745559 | MR 3292595
[35] Shaked, M., Shanthikumar, J. G.: Stochastic Orders and Their Applications. Academic Press, San Diego 2007. MR 1278322
[36] Shannon, C. E.: A mathematical theory of communication. Bell. Syst. Tech. J. 27 (1948), 379-423 and 623-656. DOI 10.1002/j.1538-7305.1948.tb01338.x | MR 0026286 | Zbl 1154.94303
[37] Sordo, M., Suarez-Llorens, A.: Stochastic comparisons of distorted variability measures. Insur. Math. Econ. 49 (2011), 11-17. DOI 10.1016/j.insmatheco.2011.01.014 | MR 2811889
[38] Toomaj, A., Crescenzo, A. Di, Doostparast, M.: Some results on information properties of coherent systems. Appl. Stoch. Mod. Bus. Ind., online first (2017). DOI 10.1002/asmb.2277
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