Some Contributions to Tolerance Intervals and Statistical Process Control
Tolerance Intervals play an important role in statistical process control along with control charts. When constructing a tolerance interval or a control chart for the mean of a quality characteristic, the normality assumption can be justifiable at least in an approximate sense. However, in applications where the individual observations are to be monitored or controlled, the normality assumption is not always satisfied. In addition, for high dimensional data, the normality is rarely, if ever, satisfied. The existing tolerance intervals for exponential random variables and sample variances are constructed under a condition that assumes a known parameter, leading to unbalanced tolerance intervals. Moreover, the existing multivariate distribution-free control charts in the literature lack the ability to identify the out-of-control variables directly from the chart signal and the scale of the original variables is often lost. In this dissertation, new tolerance intervals for exponential random variables and for the sample variances, and a multivariate distribution-free control chart are developed. This dissertation consists of three chapters. The summary of each chapter is provided below. In the first chapter, we introduce a tolerance interval for exponential random variables that gives the practitioner control over the ratio of the two tails probabilities without assuming that the parameter of the distribution, the mean, is known. The second chapter develops a tolerance interval and a guaranteed performance control chart for the sample variances without assuming that the population variance is known. The third chapter introduces a multivariate distribution-free control chart based on order statistics that can identify out-of-control variables and preserve the original scale.