Reliability estimation and order-restricted inference based on joint type-II progressive censoring scheme with application to splashing data in atomization process

Censoring commonly occurs in real-world scenarios, either intentionally or unintentionally. In addition, in engineering and applied studies, two or more populations are required to perform the experiment. In this article, we develop the order-restricted (OR) statistical inference of two inverted exponentiated Rayleigh (IERayleigh) populations based on the joint type-II progressive censoring scheme (J-IIProgCS). We also consider the problem of the stress-strength reliability Re = P(X>Y) estimation under J-IIProgCS. The process of estimating the parameters with order restriction (PaR-OR) and Re are performed using the maximum likelihood (ML) and Bayesian methods. On the presumption that the gamma priors are independent, the Bayesian estimate of reliability model is obtained using the hybrid Gibbs within Metropolis-Hastings (G-MeHa) algorithm. Moreover, with assumption the ordered Beta-Gamma prior for the shape parameters and the Gamma prior for the common scale parameter, the Bayes estimates of the parameters with order restriction are evaluated. The classical confidence intervals and the Bayesian credible interval are created. A detailed numerical illustration using simulated data sets is carried out. A real engineering example related to splashing phenomenon is considered for illustrative purposes.