On Inference of Cpyk$\mathcal {C}_{pyk}$ for Progressively Censored Log–Log Distributed Samples

ABSTRACT In this article, we focus on the estimation of the generalized process capability index, , for the Log–Log distribution under a progressive type II censoring scheme. To facilitate robust inference, we employ three distinct estimation methodologies, including maximum likelihood, maximum product of spacings, and Bayesian approaches. Within the Bayesian framework, estimators of are derived under both the squared error loss function and the LEL function, assuming independent gamma priors for the unknown parameters. Furthermore, approximate confidence intervals for the index are constructed using conventional frequentist methods and are compared with the Bayesian HPD intervals. To comprehensively evaluate the efficacy of the proposed estimation procedures, a comprehensive Monte Carlo simulation is carried out. The accuracy of the point estimators is measured by their mean squared errors and biases, whereas the reliability of the interval estimates is judged by their average widths along with coverage probabilities. The practical applicability of the proposed index and estimation methodologies is further illustrated through a real data example concerning the failure times of transmission components in Caterpillar tractors.