Estimation and Confidence Intervals of a Quantile‐Based Process Capability Index for the Power Rayleigh Distribution With ROC Analysis and Applications to Industrial Data

Process capability indices () are essential tools for assessing how well a process aligns with predefined quality standards. While larger values are often associated with higher process performance, they do not necessarily guarantee lower rejection rates. This work introduces a new formulated through the quantile function, particularly suited for processes that deviate from normality. The investigation centers on processes described by the power Rayleigh distribution with unknown parameters and pursues four main objectives. First, six distinct frequentist estimation techniques are proposed for the new PCI under this distributional framework. Second, the efficiency of three parametric bootstrap confidence intervals () is examined in estimating the index. Third, Bayesian estimation is carried out under two alternative loss functions ‐ the squared error loss and the linear exponential loss assuming gamma priors for the parameters. Interval estimation is addressed through approximate and bootstrap‐based confidence intervals in the frequentist setting and through highest posterior density credible intervals in the Bayesian setting. Fourth, the statistical power of the is investigated via the equivalence between confidence interval construction and two‐sided hypothesis testing. Their discriminative capacity is further explored using receiver operating characteristic curve analysis. Finally, a detailed Monte Carlo simulation is designed to evaluate the competing estimators with respect to mean squared error, coverage probability, and average interval length. To highlight the practical value of the proposed approaches, two case studies based on real data from the electronic and food industries are presented.