Reliability analysis under extreme conditions for the Burr XII distribution utilizing upper record values

Abstract

This article examines the reliability estimation scenario of $\omega = P(L\lt Y\lt T),$ where the strength $Y$ occurs between two extreme conditions, namely the upper extreme ($T$) and the lower extreme ($L$). Assuming that the random variables $L, T,$ and $Y$ follow a Burr XII distribution, the statistical inference of $\omega$ is examined under the upper values of the record. Maximum likelihood and parametric bootstrapping approaches are used to obtain point and confidence interval estimates of $\omega$. This study considers the stress-strength reliability estimator with uniform and gamma priors under several loss functions. Based on the proposed loss functions, reliability $\omega$ is estimated using Bayesian analyzes with Gibbs and Metropolis-Hastings samplers. In addition, we construct credible intervals that contain the highest posterior densities. Monte Carlo simulation studies and examples based on real-data are also performed to analyze the behavior of the proposed estimators. This study involves the examination of specimens of an electrically insulating fluid, especially those utilized in transformers, by applying the stress-strength model for data set analysis. Based on the study's results, it was clear that mean squared errors decreased as record numbers increased. Bayesian estimates under the precautionary loss function are commonly found to be more suitable for determining simulation conclusions than other specified loss functions.

Keywords

• Bayesian estimation
• bootstrap confidence interval
• Burr XII distribution
• Metropolis-Hastings sampler
• precautionary loss function
• record values

References

1. [1] Z.W. Birnbaum and R.C. McCarty, A Distribution Free upper confidence bound for $P(Y \lt X)$, based on independent samples of X and Y, Ann. Math. Stat. 29 (2), 558–562, 1958.
2. [2] S. Kotz, Y. Lumelskii and M. Pensky, The Stress-Strength Model and its Generalizations, World Scientific, 1–10, 2003.
3. [3] D. Kundu and R.D. Gupta, Estimation of $P(Y \lt X)$ for Weibull distribution, IEEE Trans. Reliab. 55 (2), 270–280, 2006.
4. [4] A.S. Hassan and D. Al-Sulami, Estimation of $P(Y \lt X)$ in the Case of Exponentiated Weibull Distribution, The Egyptian Statistical Journal 52 (2), 76-95, 2008.
5. [5] S. Moheb, A. S. Hassan and L.S. Diab, Classical and Bayesian inferences of stressstrength reliability model based on record data, Commun. Stat. Appl. Methods 31, 497–519, 2024.
6. [6] S.A. Alyami, A.S. Hassan, I. Elbatal, O. Albalawi, M. Elgarhy and A.R. El-Saeed, Bayesian and non-Bayesian analysis for stress-strength model based on progressively first failure censoring with applications, PLoS ONE 19 (12), 2024.
7. [7] S. Chandra and D.B. Owen, On estimating the reliability of a component subject to several different stresses (strengths), Nav. Res. Logist. Q. 22 (1), 31–39, 1975.
8. [8] N. Singh, On the estimation of $Pr(X_1 \lt Y \lt X_2)$, Commun. Stat. - Theory Methods 9 (15), 1551–1561, 1980.

9. [9] V.V. Ivshin, On the estimation of the probabilities of a double linear inequality in the case of uniform and two-parameter exponential distributions, J. Math. Sci. 88, 819–827, 1998.
10. [10] A.S. Hassan, E.A. Elsherpieny and R.M. Shalaby, On the Estimation of $P(Y \lt X \lt Z)$ for Weibull Distribution in the Presence of k Outliers, Int. J. Eng. Res. Appl. 3, 1728 – 1734, 2013.
11. [11] N. Choudhary, A. Tyagi and B. Singh, Estimation of $R = P(Y \lt X \lt Z)$ under progressive type-II censored data from Weibull distribution, Lobachevskii J. Math. 42, 318-335, 2021.
12. [12] M.M. Yousef, A.S. Hassan, H.M. Alshanbari, A-A.H. El-Bagoury and E.M. Almetwally, Bayesian and non-Bayesian analysis of exponentiated exponential stressstrength model based on generalized progressive hybrid censoring process, Axioms 11 (9), 455, 2022.
13. [13] R. Alotaibi, E.M. Almetwally, I. Ghosh and H. Rezk, The optimal multistress– strength reliability technique for the progressive first failure in the length-bias exponential model using Bayesian and non-Bayesian methods. J. Stat. Comput. Simul. 94 (11), 2436–2461, 2024.
14. [14] A.S. Nayal, B. Singh, V. Tripathi and A. Tyagi, Analyzing stress-strength reliability $\delta =P(U\lt V \lt W)$: a Bayesian and frequentist perspective with Burr-XII distribution under progressive Type-II censoring, Int. J. Syst. Assur. Eng. Manag. 15, 2453-2472, 2024a.
15. [15] A.S. Nayal, A. Tyagi, S. Rathi and B. Singh, Inferences on stress-strength reliability $R = P(Y \lt X \lt Z)$ for progressively type-II censored Weibull half logistic distribution, OPSEARCH 62, 797–832, 2024.
16. [16] A.S. Hassan, N. Alsadat, M. Elgarhy, H. Ahmad, and H.F. Nagy, On estimating multi-stress-strength reliability for inverted Kumaraswamy under ranked set sampling with application in engineering, J. Nonlinear Math. Phys. 31, 2024.
17. [17] A.S. Hassan, E.A. El-Sherpieny and R.E. Mohamed, Bayesian estimation of stress strength modeling using MC-MC method based on outliers, Ann. Data Sci. 12, 23–62, 2025.
18. [18] A.S. Hassan and Y.S. Morgan, Bayesian and non-Bayesian analysis of $R = Pr(W \lt Q \lt Z)$ for inverted Kumaraswamy distribution containing outliers with data application, Qual. Quant. 59, 3271–3303, 2025.
19. [19] N. Jacob and A. EJ, Shrinkage estimation of stress strength reliability P(Y < X < Z) for Lomax distribution based on records, Int. j. stat. Appl. Math. 8 (1), 107 - 123, 2023.
20. [20] S. Moheb, A.S. Hassan and L.S. Diab, Inference of $P(Y \lt X \lt Z)$ for Unit Exponentiated Half Logistic Distribution with Upper Record Ranked Set Samples, Sankhya A 87, 643 - 698, 2025.
21. [21] K.N. Chandler, The distribution and frequency of record values, J. R. Stat. Soc. Ser. B Methodol. 14 (2), 220 - 228, 1952.
22. [22] M. Ahsanullah, Introduction to Record Statistics, NOVA Science Publishers Inc. Huntington, 1995.
23. [23] B.C. Arnold, N. Balakrishnan and H.N. Nagaraja, Records, John Wiley & Sons, New York, 1998.
24. [24] J. Ahmadi and M. Doostparast, Bayesian estimation and prediction for some life distributions based on record values, Statistical Papers 47, 373 – 392, 2006.
25. [25] A.A. Essam, On Record Value of the Power Lomax Distribution, Adv. Appl. Stat. 50 (6), 497 – 515, 2017.
26. [26] A.S. Hassan, M. Abd-Allah and H.F. Nagy, Estimation of $P(Y \lt X)$ using record values from the generalized inverted exponential distribution, Pak. J. Stat. Oper. Res. 14, 645 - 660, 2018a.
27. [27] A.S. Hassan, M. Abd-Allah and H. F. Nagy, Bayesian analysis of record statistics based on generalized inverted exponential model, Int. J. Adv. Sci. Eng. Inf. Techno. 8 (2), 323–335, 2018b.
28. [28] I.W. Burr, Cumulative Frequency Functions, Ann. Math. Stat. 13 (2), 215 - 232, 1942.
29. [29] W.J. Zimmer, J.B. Keats and F. Wang, The Burr XII distribution in reliability analysis, J. Qual. Technol. 30 (4), 386 - 394, 1998.
30. [30] A.A. Soliman, Estimation of parameters of life from progressively censored data using Burr-XII model, IEEE Trans. Reliab. 54 (1), 34 - 42, 2005.
31. [31] Y.L. Lio and T.R. Tsai, Estimation of $\delta = P(X \lt Y )$ for Burr XII distribution based on the progressively first failure-censored samples, J. Appl. Stat. 39 (2), 309 - 322, 2012.
32. [32] J. Jia, Z. Yan and X. Peng, Parameters estimation of Burr XII distribution under first failure progressively unified hybrid censoring schemes, Stat. Anal. Data Min. 11 (6), 271 - 281, 2018.
33. [33] S. Saini, S. Tomer and R. Garg, On the reliability estimation of multicomponent stress–strength model for Burr XII distribution using progressively first-failure censored samples, J. Stat. Comput. Simul. 92 (4), 667 - 704, 2021.
34. [34] S. Dey, S. Singh, Y.M. Tripathi and A. Asgharzadeh, Estimation and prediction for a progressively censored generalized inverted exponential distribution, Stat. Methodol. 32, 185-202, 2016.
35. [35] B. Efron, The Jackknife, the Bootstrap, and Other Resampling Plans, SIAM, Philadelphia, 1987.
36. [36] B. Efron and R. Tibshirani, An Introduction to the Bootstrap, Chapman and Hall/CRC, New York, 1994.
37. [37] J. Shao and D. Tu, The Jackknife and Bootstrap, Springer, New York, N.Y., 1995.
38. [38] M. Chen and Q. Shao, Monte Carlo estimation of Bayesian credible and HPD intervals, J. Comput. Graph. Stat. 8 (1), 69 - 92, 1999.
39. [39] S. Dey, T. Dey and D. J. Luckett, Statistical inference for the generalized inverted exponential distribution based on upper record values, Math. Comput. Simul. 120, 64 - 78, 2016.
40. [40] S. Singh and Y.M. Tripathi, Estimating the parameters of an inverse Weibull distribution under progressive type-I interval censoring , Statistical Papers 59, 21 - 56, 2018.
41. [41] W. Nelson, Graphical analysis of accelerated life test data with the inverse power law model, IEEE Trans. Reliab. R-21 (1), 2 - 11, 1972.
42. [42] J.F. Lawless, Statistical Models and Methods for Lifetime Data, 2nd edition, Hoboken, New Jersey, John Wiley & Sons, 2003.
43. [43] K. Krishnamoorthy and Y. Lin, Confidence limits for stress–strength reliability involving Weibull models, J. Stat. Plan. Infer. 140 (7), 1754 - 1764, 2010.