On residual cumulative generalized exponential entropy and its application in human health

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

Numerous adaptations of traditional entropy concepts and their residual counterparts have emerged in statistical research. While some methodologies incorporate supplementary variables or reshape foundational assumptions, many ultimately align with conventional formulations. This study introduces a novel extension termed residual cumulative generalized exponential entropy to broaden the scope of residual cumulative entropy for continuous distributions. Key attributes of the proposed measure include non-negativity, bounds, its relationship to the continuous entropy measure, and stochastic comparisons. Practical implementations are demonstrated through case studies involving established probability models. Additionally, insights into order statistics are derived to characterize the measure’s theoretical underpinnings. The residual cumulative generalized exponential entropy framework bridges concepts such as Bayesian risk assessment and excess wealth ordering. For empirical implementation, non-parametric estimation strategies are devised using data-driven approximations of residual cumulative generalized exponential entropy, with two distinct estimators of the cumulative distribution function evaluated. A practical application is showcased, using clinical diabetes data. The study further explores the role of generalized exponential entropy in identifying distributional symmetry, mainly through its application to uniform distributions to pinpoint symmetry thresholds in ordered data. Finally, the utility of generalized exponential entropy is examined in pattern analysis, with a diabetes dataset serving as a benchmark for evaluating its classification performance.

Keywords

exponential entropy / order statistics / non-parametric estimation / residual cumulative entropy / pattern recognition / stochastic order

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] C. Shannon, A mathematical theory of communication, Bell Syst. Tech. J., 27 (1948), 379-423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x

[2] M. Rao, Y. Chen, B. Vemuri, F. Wang, Cumulative residual entropy: A new measure of information, IEEE Trans. Inf. Theory, 50 (2004), 1220-1228. https://doi.org/10.1109/TIT.2004.828057

[3] L. L. Campbell, Exponential entropy as a measure of extent of distribution, Z. Wahrscheinlichkeitstheorie verw Gebiete, 5 (1966), 217-225. https://doi.org/10.1007/BF00533058

[4] N. R. Pal, S. K. Pal, Object background segmentation using new definitions of entropy, IEE Proc. E, 136 (1989), 284-295. https://doi.org/10.1049/ip-e.1989.0039

[5] N. R. Pal, S. K. Pal, Entropy: A new definition and its applications, IEEE Trans. Syst. Man Cybern., 21 (1991), 1260-1270. https://doi.org/10.1109/21.120079

[6] S. M. Panjehkeh, G. R. M. Borzadaran, M. Amini, Results related to exponential entropy, Int. J. Inf. Coding Theory, 4 (2017), 258-275. https://doi.org/10.1504/IJICOT.2017.086915

[7] T. O. Kvalseth, On exponential entropies, in Proceedings of the IEEE International Conference on Systems, Man and Cybernatics, 4 (2000), 2822-2826. https://doi.org/10.1109/ICSMC.2000.884425

[8] S. S. Alotaibi, A. Elaraby, Generalized exponential fuzzy entropy approach for automatic segmentation of chest CT with COVID-19 infection, Complexity, 2022 (2022), 7541447. https://doi.org/10.1155/2022/7541447

[9] A. P. Wei, D. F. Li, B. Q. Jiang, P. P. Lin, The novel generalized exponential entropy for intuitionistic fuzzy sets and interval valued intuitionistic fuzzy sets, Int. J. Fuzzy Syst., 21 (2019), 2327-2339. https://doi.org/10.1007/s40815-019-00743-6

[10] S. Kumar, A new exponential knowledge and similarity measure with application in multi-criteria decision-making, Decis. Anal. J., 10 (2024), 100407. https://doi.org/10.1016/j.dajour.2024.100407

[11] C. Wang, G. Shi, Y. Sheng, H. Ahmadzade, Exponential entropy of uncertain sets and its applications to learning curve and portfolio optimization, J. Ind. Manage. Optim., 21 (2025), 1488-1502. https://doi.org/10.3934/jimo.2024134

[12] J. Ye, W. Cui, Exponential entropy for simplified neutrosophic sets and its application in decision making, Entropy, 20 (2018), 357. https://doi.org/10.3390/e20050357

[13] M. Shaked, J. G. Shanthikumar, Stochastic Orders and Their Applications, San Diego, Academic Press, 1994.

[14] J. S. Hwang, G. D. Lin, On a generalized moment problem. II, Proc. Am. Math. Soc., 91 (1984), 577-580. https://doi.org/10.1090/S0002-9939-1984-0746093-4

[15] U. Kamps, 10 Characterizations of distributions by recurrence relations and identities for moments of order statistics, in Order Statistics: Theory & Methods, Elsevier, 16 (1998), 291-311. https://doi.org/10.1016/S0169-7161(98)16012-1

[16] J. R. Higgins, Completeness and Basis Properties of Sets of Special Functions, Cambridge University Press, 2004.

[17] J. Galambos, The asymptotic theory of extreme order statistics, Technometric, 32 (1990), 110-111. https://doi.org/10.1080/00401706.1990.10484616

[18] G. Psarrakos, A. Toomaj, On the generalized cumulative residual entropy with applications in actuarial science, J. Comput. Appl. Math., 309 (2017), 186-199. https://doi.org/10.1016/j.cam.2016.06.037

[19] M. Asadi, N. Ebrahimi, E. S. Soofi, Connections of Gini, Fisher, and Shannon by bayes risk under proportional hazards, J. Appl. Probab., 54 (2019), 1027-1050. https://doi.org/10.1017/jpr.2017.51

[20] J. M. Fernandez-Ponce, S. C. Kochar, J. Muñoz-Perez, Partial orderings of distributions based on right spread functions, J. Appl. Probab., 35 (1998), 221-228.

[21] A. Di Crescenzo, M. Longobardi, On cumulative entropies, J. Stat. Plann. Inference, 139 (2009), 4072-4087. https://doi.org/10.1016/j.jspi.2009.05.038

[22] H. Xiong, P. Shang, Y. Zhang, Fractional cumulative residual entropy, Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 104879. http://dx.doi.org/10.1016/j.cnsns.2019.104879

[23] P. Billingsle, Probability and Measure, John Wiley & Sons, 2008.

[24] V. Zardasht, Testing the dilation order by using cumulative residual Tsallis entropy, J. Stat. Comput. Simul., 89 (2019), 1516-1525. https://doi.org/10.1080/00949655.2019.1588270

[25] B. Arnold, N. Balakrishnnan, H. N. Nagaraja, A First Course in Order Statistics, Society for Industrial and Applied Mathematics, 2008.

[26] G. M. Reaven, R. G. Miller, An attempt to define the nature of chemical diabetes using a multidimensional analysis, Diabetologia, 16 (1979), 17-24. https://doi.org/10.1007/BF00423145

[27] M. Fashandi, J. Ahmadi, Characterizations of symmetric distributions based on Renyi entropy, Stat. Probabil. Lett., 82 (2012), 798-804. https://doi.org/10.1016/j.spl.2012.01.004

[28] N. Balakrishnan, A. Selvitella, Symmetry of a distribution via symmetry of order statistics, Stat. Probabil. Lett., 129 (2017), 367-372. https://doi.org/10.1016/j.spl.2017.06.023

[29] N. Balakrishnan, F. Buono, M. Longobardi, On Tsallis extropy with an application to pattern recognition, Stat. Probab. Lett., 180 (2022), 109241. https://doi.org/10.1016/j.spl.2021.109241

[30] R. A. Aldallal, H. M. Barakat, M. S. Mohamed, Exploring weighted Tsallis extropy: Insights and applications to human health, AIMS Math., 10 (2025), 2191-2222. https://doi.org/10.3934/math.2025102

[31] B. Y. Kang, Y. Li, Y. Deng, Y. J. Zhang, X. Y. Deng, Determination of basic probability assignment based on interval numbers and its application, Acta Electron. Sin., 40 (2012), 1092-1096. https://doi.org/10.3969/j.issn.0372-2112.2012.06.004