Statistical inferences and sample size determination for process capability indices under additive and multiplicative adjustment of process mean

Abstract

Process capability analysis is concerned with evaluating the ability of a process to produce products or services that meet specifications. Such an ability is usually measured by process capability indices (PCIs). These PCIs can be classified into potential PCIs and performance PCIs. In this thesis, the so called estimation-error-based (EEB) approach is developed for determining the sample size for a precise estimation of the potential and performance PCIs. In addition, a potential PCI for the case of the multiplicative adjustment of the process mean, a new PCI called $C_{pa}$ is developed for the nominal-the-best quality characteristic, and its statistical properties are investigated. This thesis first develops the EEB approach for determining the sample size for $C_{p}$, $C_{pk}$, and $C_{pm}$. The EEB approach for $C_{p}$ allows the sample size to be determined without any knowledge of the true value of the unknown parameter. In the case of $C_{pk}$ and $C_{pm}$, additional assumptions on the unknown parameters are required. A specific method for determining these unknown parameters as well as a method for using them to obtain the sample size is presented. In addition, an approximate EEB approach based on the approximate distribution of the estimator of a PCI is developed. The approximate EEB approach provides a closed-form formula for the case of $C_{pk}$ and computationally efficient algorithm for the case of $C_{p}$ and $C_{pm}$. Secondly, this thesis investigates statistical properties of the natural estimator $C_{pa}$ of $C_{pa}$, and develops the uniformly minimum variance unbiased estimator of $C_{pa}$. In addition, the exact and approximate lower confidence limits of $C_{pa}$ are derived. As an application of the interval estimation of $C_{pa}$, process capability demonstration procedures are developed. Specifically, the exact and approximate methods for determining the minimum sample size and the minimum $C_{pa}$ value that must be obtained are de...