Order-restricted Inferences on P (X1 < Y < X2) for the Weibull Distribution Under Joint Progressive Censoring

Abstract

Estimation of $\Omega=P(X_1<Y<X_2)$ denotes reliability where the strength $Y$ should not only be greater than the stress $X_1$ but also be smaller than the stress $X_2$ and used as an important case of studies of stress-strength reliability. In this paper, the Weibull distributions with common shape parameters and different scale parameters are assumed to be the underlying distributions of the components. These three components are assumed to be performed under the joint progressive type-II censoring scheme introduced by Balakrishnan et al. (2015). In addition, a natural constraint on the scale parameters such as $\lambda_1<\lambda_2<\lambda_3$ is considered. Thus, the inference of $\Omega$ is obtained under jointly progressive censored data under order-restricted scale parameters. Maximum likelihood estimations are obtained from the findings of the generalized isotonic regression problem defined by Barlow et al. (1972).  Additionally, the Bayesian estimation is obtained under a gamma-Dirichlet prior distribution by performing the importance sampling algorithm. The asymptotic confidence intervals and the highest posterior density intervals are derived as the approximate confidence intervals. Performances of the estimation methods are evaluated by simulation studies.