The use of Cornish-Fisher expansion and nonparametric bootstrap confidence intervals for process capability indices

C{dollar}sb{lcub}p{rcub}{dollar} and C{dollar}sb{lcub}pk{rcub}{dollar} process capability indices are extensively used in industry to serve as a common language for communicating quality requirements and actual process performance relative to predetermined engineering specifications. Their use is justified only when it is evident that the process is running under a state of statistical process control and that the process is normally distributed. To enhance the meaning of these indices, several extensions--C{dollar}sb{lcub}pm{rcub}{dollar} and C{dollar}sb{lcub}pmk{rcub}{dollar}--have been proposed to properly take into account that process performance should be evaluated with respect to deviation from its target characteristic. In this research, we are concerned with issues related to the possible lack of normality of a process characteristic under study. When the data is non-normal, we advocate the use of extended capability indices which replaces the quantity 6{dollar}sigma{dollar} by the distance, {dollar}rm Xsb{lcub}99.865%{rcub}-Xsb{lcub}0.135%{rcub}{dollar}. We introduce a simple new method based upon the use of a Cornish-Fisher expansion to estimate the required percentiles, and we compare the performance of this method to other methods using well established data sets. The distribution of capability indices is highly complicated, even in the case of normally distributed process. Nonparametric bootstrap procedures have been used successfully to establish good confidence intervals estimates of the capability indices regardless of the distributional aspect of the data. The accelerated biased-corrected bootstrap confidence interval, an approximation to this interval, and an approximation to the standard bootstrap interval that do not require Monte Carlo simulation will be introduced and their accuracy is compared to other bootstrap intervals for capability indices using simulation methods.