Applications of Mathematics, first online, pp. 1-22
On stochastic properties of past varentropy with applications
Akash Sharma, Chanchal Kundu
Received August 10, 2023. Published online March 11, 2024.
Abstract: To have accuracy in the extracted information is the goal of the reliability theory investigation. In information theory, varentropy has recently been proposed to describe and measure the degree of information dispersion around entropy. Theoretical investigation on varentropy of past life has been initiated, however details on its stochastic properties are yet to be discovered. In this paper, we propose a novel stochastic order and introduce new classes of life distributions based on past varentropy. Further, we illustrate some of its applications in reliability modeling and in the diversity measure of Boltzmann distribution.
Keywords: ageing classes; past varentropy order; stochastic orders; varentropy
References: [1] E. Arikan: Varentropy decreases under polar transform. IEEE Trans. Inf. Theory 62 (2016), 3390-3400. DOI 10.1109/TIT.2016.2555841 | MR 3506740 | Zbl 1359.94292
[2] R. E. Barlow, F. Proschan: Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York (1975). MR 0438625 | Zbl 0379.62080
[3] S. Bobkov, M. Madiman: Concentration of the information in data with log-concave distributions. Ann. Probab. 39 (2011), 1528-1543. DOI 10.1214/10-AOP592 | MR 2857249 | Zbl 1227.60043
[4] F. Buono, M. Longobardi, F. Pellerey: Varentropy of past lifetimes. Math. Methods Stat. 31 (2022), 57-73. DOI 10.3103/S106653072202003X | MR 4491117 | Zbl 07595983
[5] K. P. Burnham, D. R. Anderson: Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. Springer, New York (2002). DOI 10.1007/b97636 | MR 1919620 | Zbl 1005.62007
[6] T. M. Cover, J. A. Thomas: Elements of Information Theory. Wiley Series in Telecommunications. John Wiley & Sons, New York (1991). DOI 10.1002/0471200611 | MR 1122806 | Zbl 0762.94001
[7] A. Di Crescenzo, M. Longobardi: Entropy-based measure of uncertainty in past lifetime distributions. J. Appl. Probab. 39 (2002), 434-440. DOI 10.1239/jap/1025131441 | MR 1908960 | Zbl 1003.62087
[8] A. Di Crescenzo, L. Paolillo: Analysis and applications of the residual varentropy of random lifetimes. Probab. Eng. Inf. Sci. 35 (2021), 680-698. DOI 10.1017/S0269964820000133 | MR 4276539 | Zbl 07620594
[9] A. Di Crescenzo, L. Paolillo, A. Suárez-Llorens: Stochastic comparisons, differential entropy and varentropy for distributions induced by probability density functions. Available at https://arxiv.org/abs/2103.11038 (2021), 21 pages. DOI 10.48550/arXiv.2103.11038
[10] N. Ebrahimi, S. N. U. A. Kirmani: Some results on ordering of survival functions through uncertainty. Stat. Probab. Lett. 29 (1996), 167-176. DOI 10.1016/0167-7152(95)00170-0 | MR 1411415 | Zbl 1007.62527
[11] M. Fradelizi, M. Madiman, L. Wang: Optimal concentration of information content for log-concave densities. High Dimensional Probability VII. Progress in Probability 71. Birkhäuser, Basel (2016), 45-60. DOI 10.1007/978-3-319-40519-3_3 | MR 3565259 | Zbl 1358.60036
[12] C. Kundu, S. Singh: On generalized interval entropy. Commun. Stat., Theory Methods 49 (2020), 1989-2007. DOI 10.1080/03610926.2019.1568480 | MR 4070896 | Zbl 1511.62012
[13] J. Liu: Information Theoretic Content and Probability: Ph.D. Thesis. University of Florida, Gainesville (2007). MR 2710407
[14] S. Maadani, G. R. Mohtashami Borzadaran, A. H. Rezaei Roknabadi: Varentropy of order statistics and some stochastic comparisons. Commun. Stat., Theory Methods 51 (2022), 6447-6460. DOI 10.1080/03610926.2020.1861299 | MR 4464492 | Zbl 07571136
[15] A. K. Nanda, P. Paul: Some properties of past entropy and their applications. Metrika 64 (2006), 47-61. DOI 10.1007/s00184-006-0030-6 | MR 2242557 | Zbl 1104.94007
[16] A. K. Nanda, P. Paul: Some results on generalized past entropy. J. Stat. Plann. Inference 136 (2006), 3659-3674. DOI 10.1016/j.jspi.2005.01.006 | MR 2284670 | Zbl 1098.94014
[17] R. Rajaram, B. Castellani, A. N. Wilson: Advancing Shannon entropy for measuring diversity in systems. Complexity 2017 (2017), Article ID 8715605, 10 pages. DOI 10.1155/2017/8715605 | Zbl 1407.94059
[18] M. Z. Raqab, H. A. Bayoud, G. Qiu: Varentropy of inactivity time of a random variable and its related applications. IMA J. Math. Control Inf. 39 (2022), 132-154. DOI 10.1093/imamci/dnab033 | MR 4388735 | Zbl 1492.94041
[19] M. Shaked, J. G. Shanthikumar: Stochastic Orders. Springer Series in Statistics. Springer, New York (2007). DOI 10.1007/978-0-387-34675-5 | MR 2265633 | Zbl 1111.62016
[20] R. Shanker, H. Fesshaye, S. Selvaraj: On modeling of lifetimes data using exponential and lindley distributions. Biometrics Biostat. Int. J. 2 (2015), 140-147. DOI 10.15406/bbij.2015.02.00042
[21] C. E. Shannon: A mathematical theory of communication. Bell Syst. Tech. J. 27 (1948), 379-423. DOI 10.1002/j.1538-7305.1948.tb01338.x | MR 0026286 | Zbl 1154.94303
[22] C. E. Shannon: A mathematical theory of communication. Bell Syst. Tech. J. 27 (1948), 623-656. DOI 10.1002/j.1538-7305.1948.tb00917.x | MR 0026286 | Zbl 1154.94303
[23] A. Sharma, C. Kundu: Varentropy of doubly truncated random variable. Probab. Eng. Inf. Sci. 37 (2023), 852-871. DOI 10.1017/S0269964822000225 | MR 4604154
[24] K.-S. Song: Rényi information, loglikelihood and an intrinsic distribution measure. J. Stat. Plann. Inference 93 (2001), 51-69. DOI 10.1016/S0378-3758(00)00169-5 | MR 1822388 | Zbl 0997.62003
[25] S. Zacks: Introduction to Reliability Analysis: Probability Models and Statistical Methods. Springer Texts in Statistics. Springer, New York (1992). DOI 10.1007/978-1-4612-2854-7 | MR 1138468 | Zbl 0743.62096
Affiliations: Akash Sharma, Chanchal Kundu (corresponding author), Department of Mathematical Sciences, Rajiv Gandhi Institute of Petroleum Technology, Jais 229 304, U.P., India, e-mail: 20bs0002@rgipt.ac.in, chanchal_kundu@yahoo.com, ckundu@rgipt.ac.in
