Estimation of the stress–strength reliability for the two-parameter bathtub-shaped lifetime distribution based on upper record values

Abstract

In this paper, the estimation of the stress–strength reliability $Pr(X>Y)$ based on upper record values is considered when $X$ and $Y$ are independent random variables from a two-parameter bathtub-shaped lifetime distribution with the same shape but different scale parameters. The maximum likelihood estimator (MLE), the approximate Bayes estimator and the exact confidence intervals of stress–strength reliability are obtained when the shape parameter is known. When the shape parameter is unknown, we obtain the MLE, the asymptotic confidence interval and some bootstrap confidence intervals of stress–strength reliability. In this case, we also apply the Gibbs sampling technique to study the Bayesian estimation of stress–strength reliability and the corresponding credible interval. A Monte Carlo simulation study is conducted to investigate and compare the performance of the different proposed methods in this paper. Finally, analysis of a real data set is presented for illustrative purposes.