Confidence intervals of the process capability index Cpc$C_{pc}$ revisited via modified bootstrap technique and ROC curves

We consider the process capability index (PCI), a widely used quality-related statistic used to assess the quality of products and performance of monitored processes in various industries. It is widely known that the conventional PCIs perform well when the quality process being monitored has a normal distribution. Unfortunately, using the indices to evaluate a non-normally distributed process often leads to inaccurate results. In this article, we consider a new PCI, Cpc$C_{pc}$, that can be used in both normal and non-normal scenarios. The objective of this article is threefold: (i) We provide a corrected form of the confidence interval for Cpc$C_{pc}$. (ii) We compare the performance of three nonparametric bootstrap confidence intervals (BCIs) for Cpc$C_{pc}$. Specifically, the standard bootstrap, percentile bootstrap, and bias-corrected percentile bootstrap. Under various distributional assumptions such as the normal, chi-square, Student t, Laplace, and two-parameter exponential distributions, the estimated coverage probabilities and average width of the confidence intervals and BCIs for Cpc$C_{pc}$ are compared. (iii) The power of the respective bootstrap approaches is evaluated by using the equivalence relation between confidence interval construction and two-sided hypothesis testing. We also provide the receiver operating characteristic curves to evaluate their performance. Finally, as an illustrative example, an actual data set related to groove dimensions (in inches) measured from a manufacturing process of ignition keys is re-analyzed to illustrate the utility of the proposed methods.