Inference of $$S^{\prime }_{pmk}$$ S pmk ′ based on bias-corrected methods of estimation for generalized exponential distribution

In this article, estimation for a new capability index $$S^{\prime }_{pmk}$$ S pmk ′ which is based on asymmetric loss function (linear-exponential) is discussed when the underlying process follows generalized exponential distribution. Various estimates of the model parameters are proposed including maximum likelihood method, bias-corrected maximum likelihood method and bootstrap bias-corrected maximum likelihood method, and subsequently the process capability index $$S^{\prime }_{pmk}$$ S pmk ′ are obtained. Through extensive simulation studies, we compare the performance of the aforementioned methods of estimation for the PCI $$S^{\prime }_{pmk}$$ S pmk ′ in terms of their absolute bias (AB) and mean squared errors (MSEs). Besides, four bootstrap methods are employed for constructing the confidence intervals for the index $$S^{\prime }_{pmk}$$ S pmk ′ by using the considered methods of estimation. Monte Carlo simulations are performed to compare the performances of the bootstrap confidence intervals (BCIs) with respect to average widths and coverage probabilities. Finally, to show the effectiveness of the proposed methods of estimation and BCIs, two published data sets related to electronic and food industries are analyzed. Simulation results showed that the bootstrap bias corrected maximum likelihood method of estimation gives the best results among other estimation methods in terms of AB and MSEs, while the two real data sets show that width of bias-corrected accelerated bootstrap interval is minimum among all other considered BCIs.