The Monitoring of Linear Profiles with a GLR Control Chart

Abstract

In this paper, we consider 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 a generalized likelihood ratio (GLR) control chart are discussed. 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 with 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. 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.