Statistical Inferences on Quadratic Loss Based Process Capability Index CQT Using Doubly Type-II Censored Data

The article investigates the parametric estimation of the process capability index CQT for the Power Rayleigh Distribution under a doubly Type-II censoring scheme. Motivated by reliability and quality control experiments where test items are removed from observation due to time, cost, and other constraints, we derive the likelihood function for the doubly Type-II censored sample and obtain point estimators of the model parameters and of CQT using maximum likelihood estimation, maximum product spacing estimation, least squares estimation, and Cramér–von Mises estimation. Approximate confidence intervals for CQT are constructed via the observed Fisher information for MLE, and their performances are evaluated through a comprehensive Monte Carlo simulation in terms of bias, mean squared error, interval width, and coverage probability. Additionally, three bootstrap confidence intervals, namely standard bootstrap (S-boot), percentile bootstrap (P-boot), and studentized bootstrap (T-boot), are constructed to further explore the interval estimation. To complement the classical approaches, Bayesian estimators of CQT are also developed under suitable gamma prior structures and squared error loss and are computed using Markov chain Monte Carlo techniques. The relative efficiency of the proposed estimators is examined under various sample sizes, censoring levels, and parameter configurations. Finally, the methodology is illustrated using real statistical quality control datasets, demonstrating that the Power Rayleigh Distribution under doubly Type-II censoring provides a flexible and effective framework for process capability assessment in censored lifetime experiments.