On the study of the stress-strength reliability in Weibull-$F$ Models

Abstract

In this paper, the problem of inferencing on the stress-strength reliability under Weibull-$F$ Models when the stress and strength systems belong to the different families of distributions from the Weibull-$F$ Model is considered. Some stochastic comparisons between the survival distribution functions of this model are obtained. Also, the asymptotic and several bootstrap confidence intervals of stress-strength reliability are studied. The efficiency of asymptotic and bootstrap confidence intervals are analyzed by simulation. The numerical example based on real-life data is displayed as an illustration.

Keywords

• Stress-strength reliability
• stochastic order
• bootstrap confidence interval

References

1. [1] F.G. Akgul, B. Senoglu and S. Acitas, Interval estimation of the system reliability for Weibull distribution based on ranked set sampling data, Hacettepe J. Math. Stat. 47 (5), 1404–1416, 2018.
2. [2] M. Alizadeh, M. Rasekhi, H.M. Yousof and G.G. Hamedani, The transmuted Weibull- G family of distributions, Hacettepe J. Math. Stat. 47 (6), 1671–1689, 2018.
3. [3] A.M. Almarashi, A. Algarni and M. Nassar,On estimation procedures of stressstrength reliability for Weibull distribution with application, PLoS ONE 15 (8), 2020.
4. [4] D.K. Al-Mutairi, M.E. Ghitany and D. Kundu, Inferences on stress-strength reliability from Lindley distributions, Commun. Stat. - Theory Methods 42 (8), 1443–1463, 2013.
5. [5] X. Bai, Y. Shi, Y. Liu and B. Liu, Reliability estimation of stress-strength model using finite mixture distributions under progressively interval censoring, J. Comput. Appl. Math. 348, 509–524, 2019.
6. [6] N. Balakrishnan and M. Kateri,On the maximum likelihood estimation of parameters of Weibull distribution based on complete and censored data, Stat. Probab. Lett. 78 (17), 2971–2975, 2008.
7. [7] M. Basikhasteh, F. Lak and M. Afshari, Bayesian estimation of stress-strength reliability for two-parameter bathtub-shaped lifetime distribution based on maximum ranked set sampling with unequal samples, J Stat Comput Simul 90 (16), 2975–2990, 2020.
8. [8] M. Bourguignon, R.B. Silva and G.M. Cordeiro, The Weibull-G family of probability distributions, Data Sci. J. 12 (1), 53–68, 2014.

9. [9] R.C.H. Cheng and M.A. Stephens,A goodness-of-fit test using Morans statistic with estimated parameters, Biometrika 76 (2), 385–392, 1989.
10. [10] B. Efron, Bootstrap methods: Another look at the jackknife, Ann. Stat. 7 (1), 1–26, 1979.
11. [11] A.S. Hassan, H.F. Nagy, H.Z. Muhammed and M.S. Saad, Estimation of multicomponent stress-strength reliability following Weibull distribution based on upper record values, J. Taibah Univ. Sci. 14 (1), 244–253, 2020.
12. [12] J.K. Jose, Estimation of stress-strength reliability using discrete phase type distribution, Commun. Stat. - Theory Methods 51 (2),368–386, 2022.
13. [13] M. Jovanović, B. Milošević, M. Obradović and Z. Vidović, Inference on reliability of stress-strength model with Peng-Yan extended Weibull distributions, Filomat 35 (6), 1927–1948, 2021.
14. [14] A. Khalifeh, E. Mahmoudi and A. Chaturvedi, Sequential fixed-accuracy confidence intervals for the stress-strength reliability parameter for the exponential distribution: two-stage sampling procedure, Comput. Stat. 35 (4), 1553–1575, 2020.
15. [15] S. Kotz, Y. Lumelskii and M. Pensky, The Stress-Strength Model and its Generalizations: Theory and Applications, World Scientific, Singapore, 2003.
16. [16] Z. Pakdaman and J. Ahmadi, Point estimation of the stress-strength reliability parameter for parallel system with independent and non-identical components, Commun. Stat. Simul. Comput. 47 (4), 1193–1203, 2018.
17. [17] Z. Pakdaman and J. Ahmadi, Stress-strength reliability for $P(X_{r:n_1}\leq Y_{k:n_2})$ in the exponential case, İstatistik Türk İstatistik Derneği Dergisi 6 (3), 92–102, 2013.
18. [18] M. Shaked and J. Shanthikumar, Stochastic Orders, Springer, New York, 2007.
19. [19] K.C. Siju, M. Kumar and M. Beer, Classical and Bayesian estimation of stressstrength reliability of a component having multiple states, Int. J. Qual. Reliab. Manag. 38 (2), 528–535, 2020.
20. [20] G. Srinivasa Rao, M. Aslam and O.H. Arif, Estimation of reliability in multicomponent stress-strength based on two parameter exponentiated Weibull Distribution, Commun. Stat. - Theory Methods 46 (15), 7495–7502, 2017.
21. [21] G. Srinivasa Rao, S. Mbwambo and A. Pak, Estimation of multicomponent stressstrength reliability from exponentiated inverse Rayleigh distribution, Int. j. stat. manag. syst. 24 (3), 499–519, 2021.
22. [22] M. Teimouri, MPS: an R package for modelling new families of distributions. arXiv preprint arXiv:1809.02959, 2018.
23. [23] M. Teimouri and S. Nadarajah, MPS: Estimating Through the Maximum Product Spacing Approach, R package version 2.3.1, URL https://CRAN.R project.org/package=MPS, 2019.
24. [24] L. Wang, K. Wu, Y.M. Tripathi and C. Lodhi, Reliability analysis of multicomponent stress-strength reliability from a bathtub-shaped distribution, Appl. Stat. 49 (1), 122– 142, 2022.
25. [25] K. Zografos and N. Balakrishnan, On families of beta-and generalized gammagenerated distributions and associated inference, Stat. Methodol. 6 (4), 344–362, 2009.