Nonparametric Confidence Limits of Quantile-Based Process Capability Indices

We propose an asymptotic nonparametric confidence interval for quantile-based process capability indices (PCIs) based on the superstructure modified from which contains the four basic PCIs, , , and , as special cases. Since the asymptotic variance of the estimator for quantile-based PCIs involves the density function of the underlying process, the existing asymptotic results cannot be used directly to construct confidence limits for PCIs. To obtain a consistent estimator for the asymptotic variance of the estimated quantile-based PCIs, in this paper, we propose to use the kernel density estimator for the underlying process. Consequently, the confidence limits for PCIs are established based on the consistent estimates. A real-life example from manufacturing engineering is used to illustrate the implementation of the proposed methods. Simulation studies are also presented in this paper to compare the two quantile estimators that are used in the definition of PCIs. 1. Introduction The process capability index (PCI) as a quality monitoring tool has been playing an increasingly important role in analyzing process quality and productivity. It has been widely applied in industrial engineering particularly in quality monitoring and improvement programs in manufacturing systems as well as reliability engineering and environmental engineering. Many PCIs have been proposed since Juran et al. [1] proposed the first PCI . Let USL and LSL be the upper and lower specification limits, , and be the target values. The process mean and standard deviation are denoted by and . V？nnman [2] proposed the following unified superstructure which includes the commonly used four basic PCIs, , , , and , where and are nonnegative constants. We can see from (1) that , , , and . Since the process mean and standard deviation-based PCIs implicitly assume the normality of the underlying process, applying such PCIs to skewed process may cause invalid results. Let be the th quantile of the process, that is, . Define to be the vector of quantiles to be used in the definition of PCIs. Chen and Pearn [3] modified V？nnman’s [2] and proposed a quantile-based PCI superstructure without assuming implicitly the normality of the underlying process as follows: where and which guarantee that the proportion of nonconforming parts of the process is at most 0.27% if the corresponding four PCIs take value greater than 1 for on target normal processes. will be used throughout this paper. Since two different types of quantile functions will be used in this paper, we will use notation meaning that the PCI