The objective of this dissertation is to investigate GLR control charts based on a sequential sampling scheme (SS GLR charts). Phase II monitoring is considered and the goal is to quickly detect a wide range of changes in the univariate normal process mean parameter and/or the variance parameter. The performance of the SS GLR charts is evaluated and design guidelines for SS GLR charts are provided so that practitioners can easily apply the SS GLR charts in applications. More specifically, the structure of this dissertation is as follows: We first develop a two-sided SS GLR chart for monitoring the mean μ of a normal process. The performance of the SS GLR chart is evaluated and compared with other control charts. The SS GLR chart has much better performance than that of the fixed sampling rate GLR chart. It is also shown that the overall performance of the SS GLR chart is better than that of the variable sampling interval (VSI) GLR chart and the variable sampling rate (VSR) CUSUM chart. The SS GLR chart has the additional advantage that it requires fewer parameters to be specified than other VSR charts. The optimal parameter choices are given, and regression equations are provided to find the limits for the SS GLR chart. If detecting one-sided shifts in μ is of interest, the above SS GLR chart can be modified to be a one-sided chart. The performance of this modified SS GLR chart is investigated. Next we develop an SS GLR chart for simultaneously monitoring the mean μ and the variance 𝜎² of a normal process. The performance and properties of this chart are evaluated. The design methodology and some illustrative examples are provided so that the SS GLR chart can be easily used in applications. The optimal parameter choices are given, and the performance of the SS GLR chart remains very good as long as the parameter choices are not too far away from the optimized choices. / Ph. D.

Generalized likelihood ratio (GLR) control charts are investigated for two types of statistical process monitoring (SPC) problems. The first part of this dissertation considers the problem of monitoring a normally distributed process variable when a special cause may produce a time varying linear drift in the mean. The design and application of a GLR control chart for drift detection is investigated. The GLR drift chart does not require specification of any tuning parameters by the practitioner, and has the advantage that, at the time of the signal, estimates of both the change point and the drift rate are immediately available. An equation is provided to accurately approximate the control limit. The performance of the GLR drift chart is compared to other control charts such as a standard CUSUM chart and a CUSCORE chart designed for drift detection. We also compare the GLR chart designed for drift detection to the GLR chart designed for sustained shift detection since both of them require only a control limit to be specified. In terms of the expected time for detection and in terms of the bias and mean squared error of the change-point estimators, the GLR drift chart has better performance for a wide range of drift rates relative to the GLR shift chart when the out-of-control process is truly a linear drift. The second part of the dissertation considers the problem of monitoring a linear functional relationship between a response variable and one or more explanatory variables (a linear profile). The design and application of GLR control charts for this problem are investigated. The likelihood ratio test of the GLR chart is generalized over the regression coefficients, the variance of the error term, and the possible change-point. The performance of the GLR chart is compared to various existing control charts. We show that the overall performance of the GLR chart is much better than other options in detecting a wide range of shift sizes. The existing control charts designed for certain shifts that may be of particular interest have several chart parameters that need to be specified by the user, which makes the design of such control charts more difficult. The GLR chart is very simple to design, as it is invariant to the choice of design matrix and the values of in-control parameters. Therefore there is only one design parameter (the control limit) that needs to be specified. Especially, the GLR chart can be constructed based on the sample size of n=1 at each sampling point, whereas other charts cannot be applied. Another advantage of the GLR chart is its built-in diagnostic aids that provide estimates of both the change-point and the values of linear profile parameters. / Ph. D.

When monitoring a binary process proportion p, it is usually assumed that the binary observations are independent. However, it is very common that the observations are correlated with p being the correlation between two successive observations. The first part of this research investigates the problem of monitoring p when the binary observations follow a first-order two-state Markov chain model with p remaining unchanged. A Markov Binary GLR (MBGLR) chart with an upper bound on the estimate of p is proposed to monitor a continuous stream of autocorrelated binary observations treating each observation as a sample of size n=1. The MBGLR chart with a large upper bound has good overall performance over a wide range of shifts. The MBGLR chart is optimized using the extra number of defectives (END) over a range of upper bounds for the MLE of p. The numerical results show that the optimized MBGLR chart has a smaller END than the optimized Markov binary CUSUM. The second part of this research develops a CUSUM-pp chart and a GLR-pp chart to monitor p and p simultaneously. The CUSUM-pp with two tuning parameters is designed to detect shifts in p and p when the shifted values are known. We apply two CUSUM-pp charts as a chart combination to detect increases in p and increases or decreases in p. The GLR-pp chart with an upper bound on the estimate of p, and an upper bound and a lower bound on the estimate of p works well when the shifts are unknown. We find that the GLR-pp chart has better overall performance. The last part of this research investigates the problem of monitoring p with p remains at the target value when the correlated binary observations are aggregated into samples with n>1. We assume that samples are independent and there is correlation between the observations in a sample. We proposed some GLR and CUSUM charts to monitor p and the performance of the charts are compared. The simulation results show MBNGLR has overall better performance than the other charts. / Ph. D.