Classical and Bayesian estimation of stress–-strength reliability with k- sample joint type-II censoring

Abstract

This study develops both classical and Bayesian estimation procedures for the stress--strength reliability parameter under a joint Type-II censoring scheme applied to $k$ independent strength samples from non-identical Weibull populations and a single Weibull stress population. Each strength population is assumed to follow the Weibull distribution with a common shape parameter $\beta$ but distinct scale parameters $\theta_i \ (i=1,2,\dots,k)$, while the stress population follows the Weibull distribution with the same shape parameter $\beta$ and scale parameter $\theta_{k+1}$. Maximum likelihood estimation are derived, and their existence and uniqueness are established. The asymptotic 95% confidence intervals are then constructed using the observed Fisher information matrix. In the Bayesian framework, independent gamma priors are specified, and estimation is carried out under the squared error loss function. Posterior computations are performed using Gibbs sampling combined with a Metropolis--Hastings step, and highest posterior density credible intervals are provided. The performance of the estimators is evaluated through extensive simulation studies, and in the special case $\beta=1$, closed form maximum likelihood estimations are derived for exponential populations. Finally, the practical applicability of the methodology is demonstrated with a real-world dataset on steel specimens.

Keywords

• Bayesian estimation
• exponential distribution
• joint type-II censoring
• stress-strength reliability
• Weibull distribution

Ethical Statement

There is no involvement of human participants and/or animals Informed consent in our submission.

References

1. [1] M. Crowder, Tests for a family of survival models based on extremes. Recent Advances in Reliability Theory, Springer, pp. 307–321, 2000.
2. [2] P. Congdon, Bayesian Statistical Modelling. John Wiley & Sons, 2007.
3. [3] S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. Readings in Computer Vision, Morgan Kaufmann, pp. 564–584, 1987.
4. [4] L. Devroye, On the use of probability inequalities in random variate generation. J. Stat. Comput. Simul. 20 (2), 91–100, 1984.
5. [5] M.H. Chen and Q.M. Shao, Monte Carlo estimation of Bayesian credible and HPD intervals. J. Comput. Graph. Stat. 8, 69–92, 1999.
6. [6] N. Balakrishnan and A. Rasouli, Exact likelihood inference for two exponential populations under joint Type-II censoring. Comput. Stat. Data Anal. 52 (5), 2725–2738, 2008.
7. [7] A.R. Shafay, N. Balakrishnan, and Y. Abdel-Aty, Bayesian inference based on a jointly type-II censored sample from two exponential populations. J. Stat. Comput. Simul. 84 (11), 2427–2440, 2014.
8. [8] Y. Abdel-Aty, Exact likelihood inference for two populations from two-parameter exponential distributions under joint Type-II censoring. Commun. Stat.-Theory Methods 46 (18), 9026–9041, 2017.

9. [9] Ç. Çetinkaya, Reliability estimation of the stress-strength model with non-identical jointly type-II censored Weibull component strengths. J. Stat. Comput. Simul. 91 14, 2917–2936, 2021.
10. [10] R. Goel and H. Krishna, Likelihood and Bayesian inference for k Lindley populations under joint type-II censoring scheme. Commun. Stat.-Simul. Comput. 51 (12), 1–16, 2021.
11. [11] A. Rasouli and N. Balakrishnan, Exact likelihood inference for two exponential populations under joint progressive type-II censoring. Commun. Stat.-Theory Methods 39 (12), 2172–2191, 2010.
12. [12] S. Mondal and D. Kundu, A new two-sample type-II progressive censoring scheme. Commun. Stat.-Theory Methods 48 (10), 2602–2618, 2019.
13. [13] S. Mondal and D. Kundu, Inference on Weibull parameters under a balanced twosample type-II progressive censoring scheme. Qual. Reliab. Eng. Int. 36 (1), 1–17, 2020.
14. [14] Ç. Çetinkaya, F. Sultana, and D. Kundu, Exact likelihood inference for two exponential populations under jointly generalized progressive hybrid censoring. J. Stat. Comput. Simul. 92, 1–25, 2022.
15. [15] F. Sultana, A. Koley, A. Pal, and D. Kundu, On two exponential populations under a joint adaptive type-II progressive censoring. Statistics 55 (6), 1328–1355, 2021.
16. [16] N. Balakrishnan and F. Su, Exact likelihood inference for k exponential populations under joint type-II censoring. Commun. Stat.-Simul. Comput. 44 (3), 591–613, 2015.
17. [17] R.A. Johnson and K.G. Mehrotra, Locally most powerful rank tests for the two-sample problem with censored data. Ann. Math. Stat. 43 (3), 823–831, 1972.
18. [18] N. Balakrishnan, F. Su, and K.-Y. Liu, Exact likelihood inference for k exponential populations under joint progressive type-II censoring. Commun. Stat.-Simul. Comput. 44 (4), 902–923, 2015.
19. [19] Ç. Çetinkaya and A.I. Genç, Stressstrength reliability estimation under the standard two-sided power distribution. Appl. Math. Model. 65, 72–88, 2019.
20. [20] S. Wellek, Basing the analysis of comparative bioavailability trials on an individualized statistical definition of equivalence. Biometrical J. 35 (1), 47–55, 1993.
21. [21] L. Schwartz and S. Wearden, A distribution-free asymptotic method of estimating, testing, and setting confidence limits for heritability. Biometrics 15 (2), 227–235, 1959.
22. [22] R.A. Johnson, Stress-strength models for reliability. Handbook of Statistics 7, 27–54, 1988.
23. [23] S. Kotz and M. Pensky, The Stress-Strength Model and Its Generalizations: Theory and Applications. World Scientific, 2003.
24. [24] V.K. Sharma, S.K. Singh, U. Singh, and V. Agiwal, The inverse Lindley distribution: a stress-strength reliability model with application to head and neck cancer data. J. Ind. Prod. Eng. 32 (3), 162–173, 2015.
25. [25] D. Kundu and R.D. Gupta, Estimation of $P(X \gt Y )$ for generalized exponential distribution. Metrika 61 (3), 291–308, 2005.
26. [26] E. Cramer, Inference for stress-strength models based on Weinman multivariate exponential samples. Commun. Stat.-Theory Methods 30 (2), 331–346, 2001.
27. [27] N. Singh, MVUE of $P(X \lt Y )$ for multivariate normal populations: An application to stress-strength models. IEEE Trans. Reliab. 30 (2), 192–193, 1981.
28. [28] 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.
29. [29] N. Turkkan and T. Pham-Gia, System stress-strength reliability: The multivariate case. IEEE Trans. Reliab. 56 (1), 115–124, 2007.
30. [30] A. Rezaei, F. Yousefzadeh, and S. Jomhoori, Estimation of stress-strength reliability for the multivariate SGPII distribution. Commun. Stat.-Theory Methods 49 (16), 3860–3881, 2020.
31. [31] A. Pak, M.Z. Raqab, M.R. Mahmoudi, S.S. Band, and A. Mosavi, Estimation of stress-strength reliability $R = P(X \gt Y )$ based onWeibull record data in the presence of inter-record times. Alex. Eng. J. 61 (3), 2130–2144, 2022.
32. [32] H. Krishna and R. Goel, Jointly type-II censored Lindley distributions. Commun. Stat.-Theory Methods 51 (1), 135–149, 2022.
33. [33] R. Goel and H. Krishna, Statistical inference for two Lindley populations under balanced joint progressive type-II censoring scheme. Comput. Stat. 37 (1), 263–286, 2022.
34. [34] H. Krishna and R. Goel, Inferences for two Lindley populations based on joint progressive type-II censored data. Commun. Stat.-Simul. Comput. 49, 1–18, 2020.
35. [35] H. Haj Ahmad and E.M. Almetwally, On statistical inference of generalized Pareto distribution with jointly progressive censored samples with binomial removal. Math. Probl. Eng. 2023 (1), 1821347, 2023.
36. [36] Y.Y. Abdelall, Statistical properties of a generalization Erlang truncated exponential distribution with applications and its bivariate extension. Comput. J. Math. Stat. Sci. 3 (2), 258–279, 2024.
37. [37] H.F. Martz, Bayesian Reliability Analysis. Wiley StatsRef: Statistics Reference Online, 2014.
38. [38] A. Gelman, J.B. Carlin, H.S. Stern, and D.B. Rubin, Bayesian Data Analysis. Chapman & Hall/CRC, 1995.
39. [39] P. Congdon, Bayesian Statistical Modelling. John Wiley & Sons, 2007.