Statistical inferences for stress–strength in the proportional hazard models based on progressive Type-II censored samples

The aim of this paper is to study the estimation of the reliability R=P(X<Y), when the available data have the form of progressively Type-II censored sample. It is supposed that X-sample and Y-sample are independent and generated from the proportional hazard rate (PHR) model with different proportionality parameters which includes several lifetime distributions such as exponential, Weibull (one parameter), Pareto and Burr type XII among others. Uniformly minimum variance unbiased estimator, maximum likelihood estimator (MLE), exact confidence interval (CI), asymptotic CI and Bayes estimator for the parameter of interest are derived. For PHR model, it has been shown that the expected width of the CIs and mean squared errors of the MLE and uniformly minimum variance unbiased estimator of R do not depend on the censoring schemes and the underlying distribution functions of X and Y. A Monte Carlo simulation study is conducted to compare the performance of the proposed estimators. Also, the use of the proposed estimators is shown in an illustrative example.