Measuring inaccuracies in the proportional hazard rate model based on extropy using a length-biased weighted residual approach

Abstract

In this paper, we consider the concept of the residual inaccuracy measure and extend it to its weighted version based on extropy. The properties of this measure are studied, and the discrimination principle is applied in the class of proportional hazard rate models. A characterization problem for the proposed weighted extropy-inaccuracy measure is studied. Some alternative expressions are provided as well as upper and lower limits and various inequalities related to the proposed measure. Non-parametric estimators based on the kernel density estimation method and empirical distribution function for the proposed measure are obtained, and the performance of the estimators are also discussed using some simulation studies. Finally, two real datasets are applied to illustrate our provided estimators. In general, our study highlights the potential of the weighted residual inaccuracy using extropy as a powerful tool to improve the quality and reliability of data analysis and modeling across various disciplines. Researchers and practitioners can benefit from incorporating this measure into their analytical toolkit to enhance the accuracy and effectiveness of their work.

Keywords

• extropy
• weighted measure of inaccuracy
• proportional hazard rate model
• kernel density estimation
• residual lifetime

References

1. [1] M. A. Aldahlan, Alpha power transformed log-logistic distribution with application to breaking stress data, Adv. Math. Phys., 2020, 1–9, 2020.
2. [2] N. M. Alfaer, A. M. Gemeay, H. M. Aljohani, and A. Z. Afify, Extended Log-Logistic Distribution: Inference and Actuarial Applications, Mathematics, 9, 1386, 2021.
3. [3] N. Balakrishnan, F. Buono, and M. Longobardi, On weighted extropies, Commun. Stat. Theory Methods, 51, 6250–6267, 2022.
4. [4] T. Bjerkedal, Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli, Amer. J. Hyg., 72, 130–148, 1960.
5. [5] A. Bowman, P. Hall, and T. Prvan, Bandwidth Selection for the Smoothing of Distribution Functions, Biometrika, 85(4), 799–808, 1998.
6. [6] T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed., John Wiley & Sons, Inc., Hoboken, New York, 2006.
7. [7] A. Di Crescenzo and M. Longobardi, Entropy-based measure of uncertainty in past lifetime distributions, J. Appl. Probab., 39, 434–440, 2002.
8. [8] A. Di Crescenzo and M. Longobardi, On weighted residual and past entropies, Sci. Math. Jpn., 64, 255–266, 2006.

9. [9] N. Ebrahimi, How to measure uncertainty in the residual life distributions, Sankhya A, 58, 48–57, 1996.
10. [10] E. Furman and R. Zitikis, Weighted premium calculation principles, Insur. Math. Econ., 42, 459–465, 2008.
11. [11] M. Hashempour, M. R. Kazemi, and S. Tahmasebi, On weighted cumulative residual extropy: characterization, estimation, and testing, Statistics, 56(3), 681–698, 2022.
12. [12] M. Hashempour and M. Mohammadi, On dynamic cumulative past inaccuracy measure based on extropy, Commun. Stat. - Theory Methods, 53(4), 1294–1311, 2024.
13. [13] M. Hashempour and M. Mohammadi, A new measure of inaccuracy for record statistics based on extropy, Probab. Eng. Inform. Sci., 38(1), 207–225, 2024.
14. [14] S. M. A. Jahanshahi, H. Zarei, and A. H. Khammar, On Cumulative Residual Extropy, Probab. Eng. Inform. Sci., 34(4), 605–625, 2020.
15. [15] S. Kayal, S. S. Madhavan, and R. Ganapathy, On dynamic generalized measures of inaccuracy, Statistica, 77(2), 133–148, 2017.
16. [16] M. R. Kazemi, M. Hashempour, and M. Longobardi, Weighted Cumulative Past Extropy and Its Inference, Entropy, 24(10), 1444, 2022. doi:10.3390/e24101444.
17. [17] D. F. Kerridge, Inaccuracy and inference, J. R. Stat. Soc. B, 23, 184–194, 1961.
18. [18] S. Kullback, Information Theory and Statistics, Wiley, New York, 1959.
19. [19] F. Lad, G. Sanfilippo, and G. Agro, Extropy: Complementary dual of entropy, Stat. Sci., 30, 40–58, 2015.
20. [20] J. L. Lebowitz, Boltzmanns entropy and times arrow, Phys. Today, 46(9), 32–38, 1993.
21. [21] E. T. Lee and J. W.Wang, Statistical Methods for Survival Data Analysis, John Wiley & Sons, Inc., New York, 2003.
22. [22] M. Mohammadi and M. Hashempour, On interval weighted cumulative residual and past extropies, Statistics, 56(5), 1029–1047, 2022.
23. [23] Z. Pakdaman and M. Hashempour, On dynamic survival past extropy properties, J. Stat. Res. Iran, 16(1), 229–244, 2019.
24. [24] Z. Pakdaman and M. Hashempour, Mixture representations of the extropy of conditional mixed systems and their information properties, Iran. J. Sci. Technol. Trans. A: Sci., 45(3), 1057–1064, 2019.
25. [25] E. Parzen, On estimation of a probability density function and mode, Ann. Math. Stat., 33(3), 1065–1076, 1962.
26. [26] G. P. Patil and J. K. Ord, On size-biased sampling and related form-invariant weighted distributions, Sankhya B, 38, 48–61, 1976.
27. [27] G. Qiu, The extropy of order statistics and record values, Stat. Probab. Lett., 120, 52–60, 2017.
28. [28] G. Qiu and K. Jia, The residual extropy of order statistics, Stat. Probab. Lett., 133, 15–22, 2018.
29. [29] G. Qiu and K. Jia, Extropy estimators with applications in testing uniformity, J. Nonparametr. Stat., 30(1), 182–196, 2018.
30. [30] G. Qiu, L. Wang, and X. Wang, On extropy properties of mixed systems, Probab. Eng. Inform. Sci., 33(3), 471–486, 2019.
31. [31] M. Rao, More on a new concept of entropy and information, J. Theor. Probab., 18, 967–981, 2005.
32. [32] S. Rezaei, S. Nadarajah, and N. Tahghighnia, A new three-parameter lifetime distribution, Statistics, 47, 835–860, 2013.
33. [33] M. M. Ristic and N. Balakrishnan, The gamma-exponentiated exponential distribution, J. Stat. Comput. Simul., 82, 1191–1206, 2012.
34. [34] M. Rosenblatt, Remarks on some nonparametric estimates of a density function, Ann. Math. Stat., 27(3), 832–837, 1956.
35. [35] E. I. A. Sathar and R. D. Nair, On dynamic survival extropy, Commun. Stat. Theory Methods, 50(6), 1295–1313, 2019.
36. [36] C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J., 27(3), 379–423, 1948.
37. [37] H. C. Taneja, V. Kumar, and R. Srivastava, A dynamic measure of inaccuracy between two residual lifetime distributions, Int. Math., 4(25), 1213–1220, 2009.