Geodesic
Further results on the generalized cumulative entropy
Kybernetika, Tome 53 (2017) no. 5, pp. 959-982

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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.
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.
DOI : 10.14736/kyb-2017-5-0959
Classification : 60E15, 62B10, 62N05
Keywords: generalized cumulative entropy; lower record values; reversed relevation transform; stochastic orders; parallel system 
Di Crescenzo, Antonio; Toomaj, Abdolsaeed. Further results on the generalized cumulative entropy. Kybernetika, Tome 53 (2017) no. 5, pp. 959-982. doi: 10.14736/kyb-2017-5-0959
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[1] Arnold, B. C., Balakrishnan, N., Nagaraja, H. N.: A First Course in Order Statistics. Wiley and Sons, New York 1992. | DOI | MR

[2] Asadi, M.: A new measure of association between random variables. | DOI

[3] Bagai, L., Kochar, S. C.: On tail ordering and comparison of failure rates. Comm. Stat. Theor. Meth. 15 (1986), 1377-1388. | DOI | MR

[4] Baratpour, S.: Characterizations based on cumulative residual entropy of first-order statistics. Comm. Stat. Theor. Meth. 39 (2010), 3645-3651. | DOI | MR

[5] Barlow, R. E., Proschan, F.: Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York 1975. | MR

[6] Bartoszewicz, J.: On a representation of weighted distributions. Stat. Prob. Lett. 79 (2009), 1690-1694. | DOI | MR

[7] Baxter, L. A.: Reliability applications of the relevation transform. Nav. Res. Logist. Q. 29 (1982), 323-330. | DOI | MR

[8] Block, H. W., Savits, T. H., Singh, H.: The reversed hazard rate function. Probab. Engrg. Inform. Sci. 12 (1998), 69-90. | DOI | MR

[9] Burkschat, M., Navarro, J.: Asymptotic behavior of the hazard rate in systems based on sequential order statistics. Metrika 77 (2014), 965-994. | DOI | MR

[10] Chandler, K. N.: The distribution and frequency of record values. J. Royal Stat. Soc. B 14 (1952), 220-228. | MR

[11] Chandra, N. K., Roy, D.: Some results on reversed hazard rate. Probab. Engrg. Inform. Sci. 15 (2001), 95-102. | DOI | MR

[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 | MR

[13] Crescenzo, A. Di, Longobardi, M.: Entropy-based measure of uncertainty in past lifetime distributions. J. Appl. Prob. 39 (2002), 434-440. | DOI | MR

[14] Crescenzo, A. Di, Longobardi, M.: On cumulative entropies. J. Stat. Plan. Infer. 139 (2009), 4072-4087. | DOI | MR

[15] Crescenzo, A. Di, Martinucci, B., Mulero, J.: A quantile-based probabilistic mean value theorem. Probab. Engrg. Inform. Sci. 30 (2016), 261-280. | DOI | MR

[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 | MR

[17] Hwang, J. S., Lin, G. D.: On a generalized moment problem II. Proc. Amer. Math. Soc. 91 (1984), 577-580. | DOI | MR

[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 | MR

[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 | MR

[20] Kayal, S.: On generalized cumulative entropies. Probab. Engrg. Inform. Sci. 30 (2016), 640-662. | DOI | MR

[21] Kayal, S.: On weighted generalized cumulative residual entropy of order $n$. Meth. Comp. Appl. Prob., online first (2017). | DOI

[22] Klein, I., Mangold, B., Doll, M.: Cumulative paired $\phi$-entropy. Entropy 18 (2016), 248. | DOI | MR

[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 | MR

[24] Li, Y., Yu, L., Hu, T.: Probability inequalities for weighted distributions. J. Stat. Plan. Infer. 142 (2012), 1272-1278. | DOI | MR

[25] Muliere, P., Parmigiani, G., Polson, N.: A note on the residual entropy function. Probab. Engrg. Inform. Sci. 7 (1993), 413-420. | DOI

[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 | MR

[27] Navarro, J., Psarrakos, G.: Characterizations based on generalized cumulative residual entropy functions. Comm. Stat. Theor. Meth. 46 (2017), 1247-1260. | DOI | MR

[28] Navarro, J., Rubio, R.: Comparisons of coherent systems with non-identically distributed components. J. Stat. Plann. Infer. 142 (2012), 1310-1319. | DOI | MR

[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 | MR

[30] Psarrakos, G., Navarro, J.: Generalized cumulative residual entropy and record values. Metrika 76 (2013), 623-640. | DOI | MR

[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 | MR

[32] Rao, M.: More on a new concept of entropy and information. J. Theoret. Probab. 18 (2005), 967-981. | DOI | MR

[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 | MR

[34] Rezaei, M., Gholizadeh, B., Izadkhah, S.: On relative reversed hazard rate order. Comm. Stat. Theor. Meth. 44 (2015), 300-308. | DOI | MR

[35] Shaked, M., Shanthikumar, J. G.: Stochastic Orders and Their Applications. Academic Press, San Diego 2007. | MR

[36] Shannon, C. E.: A mathematical theory of communication. Bell. Syst. Tech. J. 27 (1948), 379-423 and 623-656. | DOI | MR | Zbl

[37] Sordo, M., Suarez-Llorens, A.: Stochastic comparisons of distorted variability measures. Insur. Math. Econ. 49 (2011), 11-17. | DOI | MR

[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

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