Multicomponent stress-strength reliability based on a right long-tailed distribution

Abstract

This article deals with the problem of reliability in a multicomponent stress-strength (MSS) model when both stress and strength variables are from inverse Kumaraswamy distribution. The reliability of the system is estimated using classical and Bayesian approaches when the common second shape parameter is known or unknown. The maximum likelihood estimation and its asymptotic confidence interval for the reliability of the system are obtained. Furthermore, two other asymptotic confidence intervals are computed based on Logit and Arcsin transformations. The uniformly minimum variance unbiased estimator for the reliability of the MSS model is obtained when the common second shape parameter is known. The Bayes estimate is obtained exactly when the second shape parameter is known and it is approximated by using the Monte Carlo Markov Chain method when the second shape parameter is unknown. The highest probability density credible interval is established using the Gibbs sampling technique. Monte Carlo simulations are implemented to compare the different proposed methods. Finally, two real data sets are presented in support of the suggested procedures.

Keywords

• Inverse Kumaraswamy distribution
• Multicomponent stress-strength model
• Right long-tailed distribution

References

1. [1] A.M. Abd AL-Fattah, A.A. El-Helbawy and G.R. Al-Dayian, Inverted Kumaraswamy distribution: Properties and estimation, Pakistan. J. Stat. Oper. Res. 33 (1), 37-61, 2017.
2. [2] M.H. Abu-Moussa and M.M.M. El-Din, On estimation and prediction for the inverted Kumaraswamy distribution based on general progressive censored samples, Pakistan. J. Stat. Oper. Res. 14 (3), 717-736, 2018.
3. [3] F.G. Akgül, Reliability estimation in multicomponent stressstrength model for Topp- Leone distribution, J. Stat. Comput. Simul. 89 (15), 2914-2929, 2019.
4. [4] F.G. Akgül, Classical and Bayesian estimation of multicomponent stressstrength reliability for exponentiated Pareto distribution, Soft Comput. 25 (14), 9185-9197, 2021.
5. [5] F.G. Akgül and B. Şenoğlu, Estimation of $P(X
6. [6] D.K. Al-Mutairi, M.E. Ghitany and D. Kundu, Inferences on stress-strength reliability from Lindley distributions, Comm. Statist. Theory Methods 42 (8), 1443-1463, 2013.
7. [7] B. Al-Zahrani and S. Basloom, Estimation of the stress-strength reliability for the Dagum distribution, J. Adv. Stat 1 (3), 157-170, 2016.
8. [8] X. Bai, Y. Shi, Y. Liu and B. Liu, Reliability inference of stressstrength model for the truncated proportional hazard rate distribution under progressively Type-II censored samples, Appl. Math. Model. 65, 377-389, 2019.

9. [9] M. Basirat, S. Baratpour and J. Ahmadi, On estimation of stressstrength parameter using record values from proportional hazard rate models, Comm. Statist. Theory Methods 45 (19), 5787-5801, 2016.
10. [10] G.K. Bhattacharyya and R.A. Johnson, Estimation of reliability in a multicomponent stress-strength model, J. Amer. Statist. Assoc. 69 (348), 966-970, 1974.
11. [11] Z.W. Birnbaum, On a use of the Mann-Whitney statistic, in: Proceedings of the 3rd Berkeley Symp. Math. Statist. Prob, 1, 13-17, 1956.
12. [12] S. Chattopadhyay, T. Chakraborty, K. Ghosh and A.K. Das, Modified Lomax model: A heavy-tailed distribution for fitting large-scale real-world complex networks, Soc. Netw. Anal. Min 11 (1), 1-24, 2021.
13. [13] M.H. Chen and Q.M. Shao, Monte Carlo estimation of Bayesian credible and HPD intervals, J. Comput. Graph. Statist. 8 (1), 69-92, 1999.
14. [14] S. Dey, J. Mazucheli and M.Z. Anis, Estimation of reliability of multicomponent stressstrength for a Kumaraswamy distribution, Comm. Statist. Theory Methods 46 (4), 1560-1572, 2017.
15. [15] B. Efron, Logistic regression, survival analysis, and the Kaplan-Meier curve, J. Amer. Statist. Assoc. 83 (402), 414-425, 1988.
16. [16] S. Foss, D. Korshunov and S. Zachary, Heavy-tailed and long-tailed distributions, An Introduction to Heavy-Tailed and Subexponential Distributions, 7-42, Springer, 2013.
17. [17] A. Gelman, J.B. Carlin, H.S. Stern and D.B. Rubin, Bayesian Data Analysis, Chapman and Hall, 2003.
18. [18] A.I. Genc, Estimation of P (X> Y) with ToppLeone distribution, J. Stat. Comput. Simul. 83 (2), 326-339, 2013.
19. [19] M.E. Ghitany, D.K. Al-Mutairi and S.M. Aboukhamseen, Estimation of the reliability of a stress-strength system from power Lindley distributions, J. Stat. Comput. Simul. 44 (1), 118-136, 2015.
20. [20] F. Jamal, M. Arslan Nasir, G. Ozel, M. Elgarhy and N. Mamode Khan, Generalized inverted Kumaraswamy generated family of distributions: Theory and applications, J. Appl. Stat. 46 (16), 2927-2944, 2019.
21. [21] M.K. Jha, S. Dey and Y.M. Tripathi, Reliability estimation in a multicomponent stressstrength based on unit-Gompertz distribution, Int. J. Qual. Reliab. Manag 37 (3), 428-450, 2019.
22. [22] M.K. Jha, Y.M. Tripathi and S. Dey, Multicomponent stress-strength reliability estimation based on unit generalized Rayleigh distribution, Int. J. Qual. Reliab. Manag 38 (10), 2048-2079, 2021.
23. [23] M. Jovanovic, B. Milosevic and M. Obradovic, Estimation of stress-strength probability in a multicomponent model based on geometric distribution, Hacet. J. Math. Stat. 49 (4), 1515-1532, 2020.
24. [24] T. Kayal, Y.M. Tripathi, S. Dey and S.J. Wu, On estimating the reliability in a multicomponent stress-strength model based on Chen distribution, Comm. Statist. Theory Methods 49 (10), 2429-2447, 2020.
25. [25] F. Kızılaslan, Classical and Bayesian estimation of reliability in a multicomponent stress-strength model based on the proportional reversed hazard rate mode, Math. Comput. Simulation 38 (2), 36-62, 2017.
26. [26] F. Kızılaslan and M. Nadar, Estimation of reliability in a multicomponent stressstrength model based on a bivariate Kumaraswamy distribution, Statist. Papers 59 (1), 307-340, 2018.
27. [27] A. Kohansal and S. Shoaee, Bayesian and classical estimation of reliability in a multicomponent stress-strength model under adaptive hybrid progressive censored data, Statist. Papers 62 (1), 309-359, 2021.
28. [28] P. Kumaraswamy, A generalized probability density function for double-bounded random processes, J. Hydrol. 46 (1-2), 79-88, 1980.
29. [29] A.O. Langlands, S.J. Pocock, G.R. Kerr and S.M. Gore, Long-term survival of patients with breast cancer: a study of the curability of the disease, Br. Med. J. 2 (6200), 1247- 1251, 1979.
30. [30] F. Louzada, P.L. Ramos and D. Nascimento, The inverse Nakagami-m distribution: a novel approach in reliability, IEEE Trans. Rel. 67 (3), 1030-1042, 2018.
31. [31] A.K. Mahto and Y.M. Tripathi, Estimation of reliability in a multicomponent stressstrength model for inverted exponentiated Rayleigh distribution under progressive censoring, OPSEARCH 57, 1043-1069, 2020.
32. [32] A.K. Mahto, Y.M. Tripathi and F. Kızılaslan, Estimation of reliability in a multicomponent stressstrength model for a general class of inverted exponentiated distributions under progressive censoring, J. Stat. Theory Pract. 14 (4), 1-35, 2020.
33. [33] R.K. Maurya and Y.M. Tripathi, Reliability estimation in a multicomponent stressstrength model for Burr XII distribution under progressive censoring, Braz. J. Probab. Stat. 34 (2), 345-369, 2020.
34. [34] H.Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Phys. A: Stat. Mech. Appl. 553, 1-14, 2020.
35. [35] S.P. Mukherjee and S.S. MAITI, Stress-strength reliability in the Weibull case, Frontiers in Reliability, World Scientific, Singapore, 4, 231-248, 1998.
36. [36] W.B. Nelson, Applied Life Data Analysis, John Wiley and Sons, 2003.
37. [37] A. Pak, N.B. Khoolenjani and M.K. Rastogi, Bayesian inference on reliability in a multicomponent stress-strength bathtub-shaped model based on record values, Pakistan. J. Stat. Oper. Res. 15 (2), 431-444, 2019.
38. [38] A. Pak, A. Kumar Gupta and N. Bagheri Khoolenjani, On reliability in a multicomponent stress-strength model with power Lindley distribution, Rev. Colombiana Estadist. 41 (2), 251-267, 2018.
39. [39] G.S. Rao, M. Aslam and D. Kundu, Burr-XII distribution parametric estimation and estimation of reliability of multicomponent stress-strength, Comm. Statist. Theory Methods 44 (23), 4953-4961, 2015.
40. [40] M. Rasekhi, M.M. Saber and H.M. Yousof, Bayesian and classical inference of reliability in multicomponent stress-strength under the generalized logistic model, Comm. Statist. Theory Methods 50 (21), 51145125, 2020.
41. [41] S. Rezaei, R.A. Noughabi and S. Nadarajah, Estimation of stress-strength reliability for the generalized Pareto distribution based on progressively censored samples, Ann. Data Sci. 2 (1), 83-101, 2015.
42. [42] J. Rojo, On the preservation of some pure-tail orderings by reliability operations, Stat. Probab. Lett. 17 (3), 189-198, 1993.
43. [43] J. Rojo, On tail categorization of probability laws, J. Amer. Statist. Assoc. 91 (433), 378-384, 1996.
44. [44] E.F. Schuster, Classification of probability laws by tail behavior, J. Amer. Statist. Assoc. 79 (388), 936-939, 1984.
45. [45] V.K. Sharma, Bayesian analysis of head and neck cancer data using generalized inverse Lindley stressstrength reliability model, Comm. Statist. Theory Methods 47 (5), 1155-1180, 2018.
46. [46] V.K. Sharma, S.K. Singh, U. Singh and F. Merovci, The generalized inverse Lindley distribution: a new inverse statistical model for the study of upside-down bathtub data, Comm. Statist. Theory Methods 45 (19), 5709-5729, 2016.
47. [47] S.K. Singh, U. Singh and M. Kumar, Bayesian estimation for Poisson-exponential model under progressive type-II censoring data with binomial removal and its application to ovarian cancer data, Comm. Statist. Simulation Comput. 45 (9), 3457-3475, 2016.
48. [48] H. Xu, S. Foss and Y. Wang, Convolution and convolution-root properties of longtailed distributions, Extremes 18 (4), 605-628, 2015.
49. [49] A.S. Yadav, M. Saha, S.K. Singh and U. Singh, Bayesian estimation of the parameter and the reliability characteristics of xgamma distribution using Type-II hybrid censored data, Life. Cycle. Reliab. Saf. Eng. 8 (1), 1-10, 2019.