Bootstrap confidence intervals of generalized process capability index Cpyk using different methods of estimation

One of the indicators for evaluating the capability of a process is the process capability index. In this article, we consider six frequentest estimation methods, namely, the method of maximum likelihood (ML), the method of least square (LS), the method of weighted least square, method of percentile (PC), the method of maximum product of spacing and method of Cramèr-von-Mises (CM) to estimate the parameters and the generalized process capability index (GPCI) $ C_{pyk} $ Cpyk for the logistic exponential (LE) distribution and in particular for exponential distribution. It is well known that confidence interval (CI) provides much more information about the population characteristic of interest than does a point estimate. Hence, next, we consider three bootstrap confidence intervals (BCIs) namely, standard bootstrap (s-boot), percentile bootstrap (p-boot) and bias-corrected percentile bootstrap ( $ BC_p $ BCp-boot) for obtaining CIs of GPCI $ C_{pyk} $ Cpyk using the aforementioned methods of estimation. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average widths of the BCIs. The performances of the estimators have been compared in terms of their mean squared error using simulated samples. Simulation results showed that the estimated coverage probabilities of the p-boot CI and the s-boot CI get closer to the nominal confidence level than those of the $ BC_p $ BCp-boot CI for both cases. Finally, three real data sets are analyzed for illustrative purposes.