Journal of Advanced Mathematical Modeling

Journal of Advanced Mathematical Modeling

Dynamic Weighted Cumulative Past Extropy: Generalization, Properties, and Applications in Reliability Analysis

Document Type : Original Paper

Authors
Morteza Mohammadi 1
Majid Hashempour 2
1 Department of Statistics, Faculty of Basic Sciences, University of Zabol, Zabol, Iran
2 Department of Statistics, Faculty of Basic Sciences, University of Hormozgan, Bandar Abbas, Iran‎
10.22055/jamm.2025.47633.2301
Abstract
This paper introduces a generalization of an uncertainty measure called the dynamic weighted cumulative past extropy. The relationship of this measure with reliability metrics such as the first and second-order conditional moments of past lifetime, weighted mean past lifetime, inverse hazard rate function, and others is studied. Also, the effect of linear transformation and monotonicity properties on the dynamic weighted cumulative past extropy is examined, and upper and lower bounds for this measure are derived. Furthermore, stochastic orders based on dynamic weighted cumulative past extropy are presented for comparing populations, and the behavior of several commonly used statistical distributions, such as exponential, Rayleigh, and Pareto, is investigated based on this uncertainty measure. Subsequently, a non-parametric empirical estimator for the dynamic weighted cumulative past extropy measure is introduced, and its asymptotic properties are studied. Finally, an index for selecting the appropriate model fit to data based on the dynamic weighted cumulative past extropy measure is proposed, and its application in analyzing a real dataset is examined.
Keywords
Weighted extropy
Cumulative past ‎extropy
Conditional moments
Weighted mean past lifetime
Subjects
[1] Asadi, M. and Zohrevand, Y., 2007. On the dynamic cumulative residual entropy. Journal of Statistical
Planning and Inference, 137(6), pp. 1931-1941. doi: 10.1016/j.jspi.2006.06.035
[2] Balakrishnan, N., Buono, F. and Longobardi, M., 2022. On weighted extropies. Communications in
Statistics-Theory and Methods, 51(18), pp. 6250-6267. doi: 10.1080/03610926.2020.1860222
[3] Belis, M. and Guiasu, S., 1968. A quantitative-qualitative measure of information in cybernetic systems (Corresp.). IEEE Transactions on Information Theory, 14(4), pp. 593-594. doi:
10.1109/TIT.1968.1054185
[4] Cover, T.M., 1999. Elements of information theory. John Wiley and Sons. doi: 10.1002/047174882X
[5] Di Crescenzo, A. and Longobardi, M., 2006. On weighted residual and past entropies. Scientiae Mathematicae Japonicae, 64(1), pp. 255-266. doi: 10.32219/isms.64.2-255
[6] Di Crescenzo, A. and Longobardi, M., 2009. On cumulative entropies. Journal of statistical planning
and inference, 139(12), pp. 4072-4087. doi: 10.1016/j.jspi.2009.05.038
[7] Hashempour, M., Mohammadi, M., Jahanshahi, S.M.A. and Khammar, A.H., 2024. Modified cumulative extropies of doubly truncated random variables. Sankhya B, 86(2), pp. 558-585. doi:
10.1007/s13571-024-00328-w
[8] Hashempour, M., Mohammadi, M. and Kamari, O., 2025. On weighted version of dynamic residual
inaccuracy measure using extropy in order statistics with applications in model selection. Statistical
Papers, 66(4), pp. 1-28. doi: 10.1007/s00362-025-01696-9
[9] Kazemi, M.R., Hashempour, M. and Longobardi, M., 2022. Weighted cumulative past extropy and its
inference. Entropy, 24(10), pp. 1444-1444. doi: 10.3390/e24101444
[10] Kamari, O. and Buono, F., 2021. On extropy of past lifetime distribution. Ricerche di Matematica,
70(1), pp. 505-515. doi: 10.1007/s11587-020-00488-7
[11] Khorashadizadeh, M., 2019. A Quantile approach to the interval Shannon entropy. Journal of Statistical Research of Iran JSRI, 15(2), pp. 317-333. doi: 10.29252/jsri.15.2.317
[12] Lad, F., Sanfilippo, G. and Agró, G., 2015. Extropy: Complementary dual of entropy. Statistical
Science, 30(1), pp. 40-58. doi: 10.1214/14-STS430
[13] Marshall, A.W., Olkin, I. and Arnold, B.C., 1979. Inequalities: theory of majorization and its applications, Academic Press, New York. doi: 10.1007/978-0-387-68276-1
[14] Misagh, F., Panahi, Y., Yari, G.H. and Shahi, R., 2011. Weighted cumulative entropy and its
estimation. In 2011 ieee international conference on quality and reliability, pp. 477-480. doi:
10.1109/ICQR.2011.6031765
[15] Mohammadi, M. and Hashempour, M., 2024. On weighted version of dynamic cumulative residual
inaccuracy measure based on extropy. Statistical Papers, 65(7), pp. 4599-4629. doi: 10.1007/s00362-
024-01568-8
[16] Mohammadi, M., Hashempour, M. and Kamari, O., 2024. On the dynamic residual measure of inaccuracy based on extropy in order statistics. Probability in the Engineering and Informational Sciences,
38(3), pp. 481-502. doi: 10.1017/S0269964823000268
[17] Nair, R.D. and Sathar, E.A., 2020. On dynamic failure extropy. Journal of the Indian Society for
Probability and Statistics, 21(1), pp. 287-313. doi: 10.1007/s41096-020-00083-x
[18] Shaked, M. and Shanthikumar, J.G. eds., 2007. Stochastic orders. Springer, New York. doi:
10.1007/978-0-387-34675-5
[19] Shannon, C.E., 1948. A mathematical theory of communication. The Bell system technical journal,
27(3), pp. 379-423. doi: 10.1002/j.1538-7305.1948.tb01338.x
 

  • Receive Date 14 August 2024
  • Revise Date 24 January 2025
  • Accept Date 14 August 2025