Bayesian estimation of the process capability index $${\mathcal {C}}_{pc}$$ C pc under type II progressive censoring scheme

The objective of this paper is to obtain the Bayesian estimates of a process capability index, denoted by $${\mathcal {C}}_{pc}$$ C pc , which is based on the proportion of conformance and is applicable to both normally as well as non-normally distributed processes, and to both continuous and discrete distributed processes. In this paper, the underlying distribution is assumed as exponentiated-exponential distribution and the sampling scheme is Type II progressive censoring. Here, seven different loss functions, namely, squared error loss function, logarithm squared error loss function, weighted squared error loss function, modified squared error loss function, entropy loss function, precautionary loss function and K-loss function to obtain the Bayes estimates of the process capability index $${\mathcal {C}}_{pc}$$ C pc using gamma prior have been considered. The Markov-Chain-Monte-Carlo simulation technique has been efficiently used here to secure an approximate solution for the considered process capability index $${\mathcal {C}}_{pc}$$ C pc . Through these extensive simulation studies and with two real life examples related to electronic and food industries, we compare the performances of the Bayes estimates based on different loss functions and the Bayes credible intervals in terms of averages widths and corresponding coverage probabilities of the considered process capability index $${\mathcal {C}}_{pc}$$ C pc . From this extensive study we observed that The PR of each estimator decreases as the effective sample size increases, for fixed sample size and censoring scheme, confirming the consistency of all estimators under different loss functions. No distinct trend is observed among Bayes estimators across censoring parameters, though differences in average posterior risks under various loss functions remain minimal. Additionally, AWs decrease with increasing n and m, while CPs converge to the nominal value.