Comparison between two generalized process capability indices for Burr XII distribution using bootstrap confidence intervals

Process capability index is an effective indicator for gauging the capability of a process potential and performance. It is used to quantify the relation between the actual performance of the process and the preset specifications of the product. In this article, we utilize bootstrap re-sampling simulation method to construct bootstrap confidence intervals, namely, standard bootstrap, percentile bootstrap, Student’s t bootstrap (STB) and bias-corrected percentile bootstrap to study the difference between two generalized process capability indices CpTk1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\text {pTk1}}$$\end{document} and CpTk2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\text {pTk2}}$$\end{document} (δ=CpTk1-CpTk2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =C_{\text {pTk1}}-C_{\text {pTk2}}$$\end{document}) and to select the better of the two processes or manufacturer’s (or supplier’s) through simulation when the underlying distribution is Burr XII distribution. The model parameters are estimated by maximum likelihood method. The proposed four bootstrap confidence intervals can be effectively employed to determine which one of the two processes or manufacturers (or suppliers) has a better process capability. Monte Carlo simulations are performed to compare the performances of the proposed bootstrap confidence intervals for δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} in terms of their estimated average widths, coverage probabilities and relative coverages. Simulation results showed that the estimated average width and relative coverages of the STB confidence interval perform better than their counterparts. Finally, real data are presented to illustrate the bootstrap confidence intervals of the difference between two process capability indices.