Abstract
We investigate the cumulative Tsallis entropy, an information measure recently introduced as a cumulative version of the classical Tsallis differential entropy, which is itself a generalization of the Boltzmann-Gibbs statistics. This functional is here considered as a perturbation of the expected mean residual life via some power weight function. This point of view leads to the introduction of the dual cumulative Tsallis entropy and of two families of coherent risk measures generalizing those built on mean residual life. We characterize the finiteness of the cumulative Tsallis entropy in terms of \({\mathcal{L}}_{p}\)-spaces and show how they determine the underlying distribution. The range of the functional is exactly described under various constraints, with optimal bounds improving on all those previously available in the literature. Whereas the maximization of the Tsallis differential entropy gives rise to the classical \(q\)-Gaussian distribution which is a generalization of the Gaussian having a finite range or heavy tails, the maximization of the cumulative Tsallis entropy leads to an analogous perturbation of the Logistic distribution.
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References
Alzer, H., Berg, C.: Some classes of completely monotonic functions, II. Ramanujan J. 11, 225–248 (2006)
Balakrishnan, N., Buono, F., On cumulative entropies in terms of moments of order statistics. Methodol. Comput. Appl. Probab. 24, 345–359 (2022)
Calì, C., Longobardi, M., Ahmadi, J.: Some properties of cumulative Tsallis entropies. Physica A 486, 1012–1021 (2017)
Charpentier, A.: Mesures de risque. In: Droesbeke, J.-J., et al. (eds.) Approches Statistiques du Risque, pp. 41–85. Technip, Paris (2014)
Di Crescenzo, A., Longobardi, M.: On cumulative entropies. J. Stat. Plan. Inference 139, 4072–4087 (2009)
Di Crescenzo, A., Toomaj, A.: Further results on the generalized cumulative entropy. Kybernetika 53(5), 959–982 (2017)
Hu, T., Chen, O.: On a family of coherent measures of variability. Insur. Math. Econ. 95, 173–182 (2020)
Kayal, S.: On generalized cumulative residual entropies. Probab. Eng. Inf. Sci. 30, 640–662 (2016)
Kayal, S.: On weighted generalized cumulative residual entropy of order \(n\). Methodol. Comput. Appl. Probab. 20, 487–503 (2018)
Krakowski, M.: The relevation transform and a generalization of the Gamma distribution function. RAIRO 7(2), 107–120 (1973)
Lau, K.-S., Prakasa Rao, B.L.S.: Characterization of the exponential distribution by the relevation transform. J. Appl. Probab. 27(3), 726–729 (1990)
Lin, G.-D.: Recent developments on the moment problem. J. Stat. Distrib. Appl. 4, 5 (2017)
Lomnicki, Z.A.: The standard error of Gini’s mean difference. Ann. Math. Stat. 23(4), 635–637 (1952)
Luo, Q.-M., Qi, F.: Bounds for the ratio of two gamma functions - from Wendel’s and related inequalities to logarithmically completely monotonic functions. Banach J. Math. Anal. 6(2), 132–158 (2012)
Melbourne, J., Nayar, P., Roberto, C.: Minimum entropy of a log-concave variable with fixed variance. Available at arXiv:2309.01840
Parsa, H., Tahmasebi, S.: Notes on cumulative entropy as a risk measure. Stoch. Qual. Control 34(1), 1–7 (2019)
Prato, D., Tsallis, C.: Nonextensive foundation of Lévy distributions. Phys. Rev. E 60, 2398 (1999)
Psarrakos, G., Navarro, J.: Generalized cumulative residual entropy and record values. Metrika 76, 623–640 (2013)
Psarrakos, G., Toomaj, A.: On the generalized cumulative residual entropy with applications in actuarial science. J. Comput. Appl. Math. 309, 186–199 (2017)
Rajesh, G., Sunoj, S.M.: Some properties of cumulative Tsallis entropy of order \(\alpha \). Stat. Pap. 60, 933–943 (2019)
Rao, M.: More on a new concept of entropy and information. J. Theor. Probab. 18, 967–981 (2005)
Rao, M., Chen, Y., Vemuri, B.C., Wang, F.: Cumulative residual entropy: a new measure of information. IEEE Trans. Inf. Theory 50, 1220–1228 (2004)
Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill, New York (1953)
Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97(2), 255–261 (1986)
Shaked, M., Shanthikumar, J.G.: Stochastic Orders and Their Applications. Academic Press, Boston (1994)
Toomaj, A., Sunoj, S.M., Navarro, J.: Some properties of the cumulative residual entropy of coherent and mixed systems. J. Appl. Probab. 54(3), 379–393 (2017)
Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988)
Wang, S., Dhaene, J.: Comonotonicity, correlation order and premium principles. Insur. Math. Econ. 22, 235–242 (1998)
Zhao, T., Wang, M.: A lower bound for the beta function. Available at arXiv:2305.02754
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Simon, T., Dulac, G. On Cumulative Tsallis Entropies. Acta Appl Math 188, 9 (2023). https://doi.org/10.1007/s10440-023-00620-3
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DOI: https://doi.org/10.1007/s10440-023-00620-3