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Some properties of cumulative Tsallis entropy. (English) Zbl 1499.94025

Summary: The cumulative entropy is an information measure which is alternative to the differential entropy. Indeed, the cumulative entropy of a random lifetime \(X\) can be expressed as the expectation of its mean inactivity time evaluated at \(X\). In this paper we propose a new generalized cumulative entropy based on Tsallis entropy (CTE) and its dynamic version (DCTE). We study some properties and characterization results for this measure.

MSC:

94A17 Measures of information, entropy
60E15 Inequalities; stochastic orderings
62N05 Reliability and life testing
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[1] Shannon, C. E., A mathematical theory of communication, Bell Syst. Tech. J., 27, 279-423 (1948) · Zbl 1154.94303
[2] Belis, M.; Guiaşu, S., A quantitative-qualitative measure of information in cybernetic systems, IEEE Trans. Inform. Theory, 4, 593-594 (1968)
[3] Di Crescenzo, A.; Longobardi, M., On weighted residual and past entropies, Sci. Math. Japan., 64, 255-266 (2006) · Zbl 1106.62114
[4] Rao, M.; Chen, Y.; Vemuri, B. C.; Wang, F., Cumulative residual entropy: a new measure of information, IEEE Trans. Inform. Theory, 50, 1220-1228 (2004) · Zbl 1302.94025
[5] Asadi, M.; Zohrevand, Y., On the dynamic cumulative residual entropy, J. Statist. Plann. Inference, 137, 1931-1941 (2007) · Zbl 1118.62006
[6] Di Crescenzo, A.; Longobardi, M., On cumulative entropies, J. Statist. Plann. Inference, 139, 4072-4087 (2009) · Zbl 1172.94543
[7] Di Crescenzo, A.; Longobardi, M., Neuronal data analysis based on the empirical cumulative entropy, (Moreno-Diaz, R.; Pichler, F.; Quesada-Arencibia, A., Computer Aided Systems Theory, EUROCAST 2011, Part I. Computer Aided Systems Theory, EUROCAST 2011, Part I, Lecture Notes in Computer Science, LNCS, vol. 6927 (2012), Springer-Verlag: Springer-Verlag Berlin, Heidelberg), 72-79
[8] Bowden, R. J., Information, measure shifts and distribution metrics, Statistics, 46, 249-262 (2010) · Zbl 1241.62013
[9] Di Crescenzo, A.; Longobardi, M., On cumulative entropies and lifetime estimations, (Mira, J.; Ferrandez, J. M.; Alvarez Sanchez, J. R.; Paz, F.; Toledo, J., Methods and Models in Artificial and Natural Computation, IWINAC 2009, Part I. Methods and Models in Artificial and Natural Computation, IWINAC 2009, Part I, LNCS, vol. 5601 (2009), Springer-Verlag: Springer-Verlag Berlin, Heidelberg), 132-141
[10] Di Crescenzo, A.; Longobardi, M., More on cumulative entropy, (Trappl, R., Cybernetics and Systems 2010 (2010), Austrian Society for Cybernetic Studies: Austrian Society for Cybernetic Studies Vienna), 181-186
[11] Di Crescenzo, A.; Longobardi, M., Stochastic comparisons of cumulative entropies, (Li, H.; Li, X., Stochastic Orders in Reliability and Risk, in Honor of Professor Moshe Shaked. Stochastic Orders in Reliability and Risk, in Honor of Professor Moshe Shaked, Lecture Notes in Statistics, vol. 208 (2013), Springer: Springer New York), 167-182 · Zbl 1312.62011
[12] Kerridge, D. F., Inaccuracy and inference, J. R. Stat. Soc. Ser. B Stat. Methodol., 23, 184-194 (1961) · Zbl 0112.10302
[13] Kullback, S.; Leibler, R. A., On information and sufficiency, Ann. Math. Stat., 22, 79-86 (1951) · Zbl 0042.38403
[14] Ahmadi, J.; Di Crescenzo, A.; Longobardi, M., On the dynamic mutual information for bivariate lifetimes, Adv. Appl. Probab., 47, 1157-1174 (2015) · Zbl 1355.94022
[15] Kundu, C.; Di Crescenzo, A.; Longobardi, M., On cumulative residual (past) inaccuracy for truncated random variables, Metrika, 76, 335-356 (2016) · Zbl 1333.94025
[16] Ebrahimi, N., How to measure uncertainty in the residual life time distribution, Sankhya A, 58, 48-56 (1996) · Zbl 0893.62098
[17] Di Crescenzo, A.; Longobardi, M., Entropy-based measure of uncertainty in past lifetime distributions, J. Appl. Probab., 39, 434-440 (2002) · Zbl 1003.62087
[18] Kayid, M.; Ahmad, I. A., On the mean inactivity time ordering with reliability applications, Probab. Eng. Inform. Sci., 18, 395-409 (2004) · Zbl 1059.62105
[19] Misra, N.; Gupta, N.; Dhariyal, I. D., Stochastic properties of residual life and inactivity time at a random time, Stoch. Models, 24, 89-102 (2008) · Zbl 1143.60055
[20] Nanda, A. K.; Singh, H.; Misra, N.; Paul, P., Reliability properties of reversed residual lifetime, Comm. Statist. Theory Methods, 32, 2031-2042 (2003), (with correction in Comm. Statist. Theory Methods 33 (2004) 991-992) · Zbl 1156.62360
[21] Barlow, R. E.; Proschan, F., Mathematical Theory of Reliability (1965), Wiley: Wiley New York · Zbl 0132.39302
[22] Gupta, R. C.; Gupta, R. D., Proportional reversed hazard rate model and its applications, J. Statist. Plann. Inference, 137, 3525-3536 (2007) · Zbl 1119.62098
[23] Longobardi, M., Cumulative measures of information and stochastic orders, Ricerche Mat., 63, S209-S223 (2014) · Zbl 1360.94163
[24] Tsallis, C., Possible generalization of Boltzmann-Gibbs statistic, J. Stat. Phys., 52, 479-487 (1988) · Zbl 1082.82501
[25] Kumar, V., Some results on Tsallis entropy measure and k-record values, Physica A, 462, 667-673 (2016) · Zbl 1400.82013
[26] Baratpour, S.; Khammar, A., Tsallis entropy properties of order statistics and some stochastic comparisons, J. Statist. Res. Iran, 13, 25-41 (2016)
[27] Zhang, Z., Uniform estimates on the Tsallis entropies, Lett. Math. Phys., 80, 171-181 (2007) · Zbl 1208.94035
[28] Sati, M. M.; Gupta, N., Some characterization results on dynamic cumulative residual Tsallis entropy, J Probab. Stat. (2015) · Zbl 1426.62016
[29] Rajesh, G.; Sunoj, S. M., Some properties of cumulative Tsallis entropy of order \(\alpha \), (S.M. Stat. Papers (2016)) · Zbl 1419.62015
[30] Wilk, G.; Wlodarczyk, Z., Examples of a possible interpretation of Tsallis entropy, Physica A, 341, 4809-4813 (2008)
[31] Forte, B.; Sastri, C. C.A., Is something missing in the Boltzmann entropy?, J. Math. Phys., 16, 1453-1456 (1975)
[32] Shaked, M.; Shanthikumar, J. G., Stochastic Orders (2007), Springer: Springer New York · Zbl 1111.62016
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