Stress-strength reliability is an important concept in reliability analysis, quantifies the probability that the strength of a system surpasses its applied stress. This paper focuses on the reliability analysis for exponentiated exponential distribution strength variable and the exponentiated Weibull distribution stress variable. The study explores the estimation of the parameters in stress-strength reliability model using maximum likelihood estimation and Bayesian estimation. In particular, the Bayesian estimator of stress-strength reliability is obtained by utilizing Lindley’s approximation by considering both linear exponential loss function and squared error loss function for informative and non-informative priors. A comprehensive simulation study is conducted and the performances of estimators are compared using mean squared errors. The stress-strength reliability for real datasets is also investigated for real-time data sets.
I would like to thank my supervisor for her continuous support and encouragement in preparing the manuscript.
The study focuses on estimating the stress-strength reliability using EED as the strength variable and EWD as the stress variable. The estimation is performed using MLE and Bayesian methods under LINEX loss function and SELF using Lindley’s approximation. The performance of these estimators is compared based on the mean squared errors. The simulation study reveals the following findings.
1) Increasing the values of while keeping other parameters fixed leads to an increase in stress-strength reliability.
2) Decreasing the values of and while keeping the remaining parameters fixed results in an increase in stress-strength reliability.
3) The Bayes estimator with a gamma prior under LLF demonstrates better performance with smaller mean squared errors across all three sets of hyperparameters.
Hence, it can be concluded that the Bayes estimators for the gamma prior under the LINEX loss function with a positive loss parameter outperform other estimation methods. Additionally, the stress-strength reliability) of the EED with EWD is investigated using real data sets of breaking strength in carbon fibers with different gauge lengths. The findings reveal that the 10 mm length carbon fibers exhibit greater strength compared to the 20 mm length fibers.
 Chaturvedi and S. Vyas, Estimation and testing procedures for the reliability functions of exponentiated distributions, Statistica, anno LXXVII, n. 1 (2017).
 A. Chaturvedi and A. Pathak, Estimating the reliability function for a family of exponentiated distributions, Journal of Probability and Statistics (2014) doi:10.1155/2014/563093
 A. Chaturvedi and A. Pathak, Estimation of the reliability function for exponentiated Weibull distribution, Journal of Statistics and Applications 7(3-4) (2012), 113-120.
 A. Chaturvedi and T. Kumari, Robust Bayesian analysis of generalized half logistic distribution, Statistics, Optimization & Information Computing 5(2) (2017), 158-178.
 A. Kohansal, LARGE estimation of the stress-strength reliability of progressively censored inverted exponentiated Rayleigh distributions, Journal of Applied Mathematics 13(1) (2017), 49-76.
 C. Li and H. Hao, Reliability of a stress-strength model with inverse Weibull distribution, IAENG International Journal of Applied Mathematics 47(3) (2017), 302-306.
 D. V. Lindley, Approximate Bayesian methods, Trabajos de Estadística Y de Investigación Operativa, 31(1) (1980), 223-245.
 F. H. Eissa, Stress-strength reliability model with the exponentiated Weibull distribution: Inferences and applications, International Journal of Statistics Probability 7(4) (2018), 78-90.
 H. A. Muttlak, W. A. Abu-Dayyeh, M. F. Saleh and E. Al-Sawi, Estimating using ranked set sampling in case of the exponential distribution, Communications in Statistics-Theory and Methods 39(10) (2010), 1855-1868.
 H. Qin, N. Jana, S. Kumar and K. Chatterjee, Stress-strength models with more than two states under exponential distribution, Communications in Statistics-Theory and Methods 46(1) (2017), 120-132.
 M. A. Hussian, Estimation of stress-strength model for generalized inverted exponential distribution using ranked set sampling, International Journal of Advances in Engineering & Technology 6(6) (2014), 2354-2362.
 N. Jana, S. Kumar and K. Chatterjee, Bayes estimation for exponential distributions with common location parameter and applications to multi-state reliability models, Journal of Applied Statistics 43(15) (2016), 2697-2712.
 N. Jana, S. Kumar, K. Chatterjee and P. Kundu, Estimating stress-strength reliability for exponential distributions with different location and scale parameters, International Journal of Advances in Engineering Sciences and Applied Mathematics 13(2-3) (2021), 177-190.
 P. Kundu, N. Jana, S. Kumar and K. Chatterjee, Stress-strength reliability estimation for exponentially distributed system with common minimum guarantee time, Communications in Statistics-Theory and Methods 49(14) (2020), 3375 3396.
 P. Kundu, S. Kumar and K. Chatterjee, Estimating the reliability function, Calcutta Statistical Association Bulletin 67(3-4) (2015), 143-161.
 P. V. Pandit and S. Joshi, Reliability estimation in multicomponent stress- strength model based on generalized Pareto distribution, American Journal of Applied Mathematics 6(5) (2018), 210-217.
 R. D. Gupta and D. Kundu, Theory and methods: Generalized exponential distributions, Australian and New Zealand Journal of Statistics 41(2) (1999), 173 188.
 S. Sengupta and S. Mukhuti, Unbiased estimation of using ranked set sample data, Statistics 42(3) (2008), 223-230.
 Z. W. Birnbaum, On a use of the Mann-Whitney statistic, In: Proceedings of the 3rd Berkley Symposium in Mathematics, Statistics and Probability 3(1) (1956), 13-17.
 Z. W. Birnbaum and R. C. McCarty, A distribution-free upper confidence bound for based on independent samples of X and Y, The Annals of Mathematical Statistics 29(2) (1958), 558-562.
 X. Li, N. Balakrishnan and N. Misra, Estimation of stress-strength reliability under different censoring schemes, Journal of Statistical Computation and Simulation 92(5) (2022), 986-1003.
 M. G. Badar and A. M. Priest, Statistical aspects of fiber and bundle strength in hybrid composites. Progress in Science and Engineering Composites. In: Hayashi, T., Kawata, K., Umekawa, S., eds. Progress in Science and Engineering Composites. Tokyo: ICCM-IV (1982), 1129-1136.