Process capability indices (PCIs) are commonly utilized for evaluating a process’s performance in meeting specified criteria. In this study, our first objective is to examine the performance of generalized process capability index (GPCI) \(\mathcal {C}_{py}\) when the quality characteristic follows some finite mixture distributions, viz., xgamma and Akash distributions. Following this, our objectives are to calculate the GPCI \(\mathcal {C}_{py}\) for quality attributes that conform to the xgamma and Akash distributions. This will be achieved by employing both the maximum likelihood estimation (MLE) and minimum variance unbiased estimation (MVUE) techniques. Subsequently, we will evaluate and compare the effectiveness of these estimation methods by examining their mean squared errors Monte-Carlo simulation study. Furthermore, we will utilize asymptotic confidence intervals (ACIs) to construct confidence intervals for the \(\mathcal {C}_{py}\) index within these distributions. To assess the effectiveness of the ACIs, we plan to analyze their average width and coverage probabilities employing Monte Carlo simulation techniques. To showcase the efficacy of the suggested methods of estimation (MLE, MVUE) and ACIs of \(\mathcal {C}_{py}\) , we have conducted data analysis based on four real data sets related to electronic and food industries.
