Estimation of Reliability under Conditional Stress - Strength Setup based on Weibull Distribution

Architha M

Department of Statistics, Bangalore University, India.

Parameshwar V Pandit *

Department of Statistics, Bangalore University, India.

*Author to whom correspondence should be addressed.


The Weibull distribution has been extensively studied and applied across various fields due to its versatility in modeling a wide range of phenomena, especially in reliability engineering, survival analysis, and lifetime modeling. The concept of Rl a,b , which represents a system's reliability in a conditional stress-strength setup, was proposed by Sabre and Khorshidian (2021). In this research, the problem of estimating reliability of the component is considered when strength variable X and stress variable Y follow independent Weibull distributions with common shapes and different scale parameters under conditional stress-strength setup. The maximum likelihood estimator, asymptotic confidence interval, Bootstrap estimators, Boot-p estimators, and Bayes estimator under-squared error loss function with associated highest posterior density interval are constructed for conditional stress-strength reliability. Simulation study is conducted to estimate mean square error (MSE) of estimator of conditional stress-strength reliability. The real data analysis is also carried out.

Keywords: Weibull distribution, stress-strength reliability, conditional stress-strength model, maximum likelihood estimator, bootstrap confidence interval Bayes estimator, MCMC technique

How to Cite

Architha M, & Pandit, P. V. (2024). Estimation of Reliability under Conditional Stress - Strength Setup based on Weibull Distribution. Asian Journal of Probability and Statistics, 26(3), 28–43.


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Chruch JD, Harris B. The estimation of Reliability from Stree-Strength relationships. Technometrics. 1970;12:49-54.

Owen DB, Craswell KJ, Hanson DL. Nonparametric upper con dence bounds for Pr(Y < X) and con dence limits for Pr(Y < X) when X and Y are normal. Journal of the American Statistical Association. 1964;9:06- 24.

Kelly GD, Kelly JA, Schucany WR. Ecient estimation of P(Y < X) in the exponential case. Technometrics. 1976;18:359-360.

Tong H. A note on the estimation of P[Y < X] in the exponential case. Technometrics. 1974;16:625.

Beg MA, Singh N. Estimation ofPr(Y < X) for the pareto distribution. IEEE Transactions on Reliability. 1979;R 28:411-414. DOI 10.1109/TR.1979.5220665

Tong H. On the estimation of Pr(Y X) for exponential families. IEEE Transactions on Reliability. 1977;R- 26:54-56.

Bilikam JE. Some stochastic stress-strength processes. IEEE Transactions on Reliability. 1985;R-34(3):269-274. DOI: 10.1109/TR.1985.5222143

Erylmaz S. Estimation in coherent reliability systems through copulas. Reliab. Eng. Syst. Saf. 2011;96(5):564{568.

Pandit PV, Joshi S. Reliability estimation in multicomponent stress-strength model based on generalized pareto distribution. American Journal of Applied Mathematics and Statistics. 2018;6(5):210-217. DOI: 10.12691/ajams-6-5-5

Saber M, Khorshidian K. Introduction to reliability for conditional stress-strength parameter. Journal of Sciences; 2021.

Riad FH, Saber M, Taghipour M, Abd El-Raouf M. Classical and bayesian inference of conditional stress-

strength model under Kumaraswamy distribution. Computational Intelligence and Neuroscience; 2021. DOI:

Saber M, Mohie El-Din MM, Yousof HM. Reliability estimation for the remained stress-strength model under the generalized exponential lifetime distribution. Journal of Probability and Statistics; 2021. DOI:

Nelson W. Weibull analysis of reliability data with few or no failures. Journal of Quality Technology. 1985;17(3):140-146. DOI: 10.1080/00224065.1985.11978953

Krishnamoorthy K, Yin Lin. Con dence limits for stress{strength reliability involving Weibull models. Journal of Statistical Planning and Inference. 2010;140(7):1754-1764.

Kundu D, Gupta RD. Estimation of P(Y < X) for Weibull distributions. IEEE Transactions on Reliability. 2006;55(2):270-280.

Efron B, Tibshirani RJ. An introduction to the bootstrap. Chapman and Hall, New York; 1993.

Chen MH, Shao QM. Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics. 1999;8:69-92.

Henningsen, Arne, Toomet, Ott. maxLik: A package for maximum likelihood estimation in R. Computational Statistics. 2011;26(3):443-458. DOI 10.1007/s00180-010-0217-1

Badar MG, Priest AM. Statistical aspects of ber and bundle strength in hybrid composites. Progress in Science and Engineering Composites. ICCM-IV, Tokyo. 1982;1129-1136.