Identifying the time of a step-change with χ² control charts

If a control chart signals a change in the process parameter, identifying the time of change will substantially help the signal diagnostics procedure because it simplifies the search for special causes. In this article, we propose a maximum likelihood estimator for the time of a step-change in a multivariate process mean when the observations follow a multivariate Normal distribution. We describe how this estimator can be used to identify the change point when a multivariate χ² control chart signals a change in the process mean. We illustrate the use of our proposed estimator with an example. We assess the performance of the estimator through computer simulation experiments. The results show that our proposed estimator performs effectively and equally well for all process dimensions and shift magnitudes considered. Thus, the estimator provides process engineers with an accurate and useful estimate of the actual time of the change in the process mean.