| We consider statistical process control (SPC) for phase II monitoring of the process mean when process measurements are multivariate. In cases when the direction of a potential mean shift is known, Healy (1987) generalized the univariate cumulative sum (CUSUM) chart to multivariate cases, using the likelihood ratio inferences, and the generalized CUSUM chart was shown to be efficient. Other existing multivariate control charts usually do not use any prior information about the potential mean shift, making them less efficient in cases when such prior information is available. We first suggest a multivariate CUSUM chart for applications in which the mean shift direction is not completely known but it follows a prior distribution. This chart can be regarded as a compromise of Healy's CUSUM chart and other existing multivariate control charts, in terms of proper accommodation of prior information about the potential shift. Numerical studies show that it performs well in cases when such prior information is available.;Conventional multivariate SPC charts, such as the Crosier's CUSUM and the multivariate EWMA charts, can only give signals of mean shift, and they cannot tell us which components of the response have shifted and what type of shift (e.g., upward or downward shift) has occurred. We second propose a CUSUM chart based on the maximum and minimum of the process response components, which is labeled as the MM-CUSUM chart and which reduces the dimension of the monitoring problem to 2. Since the maximum of the response components is sensitive to upward mean shifts, and the minimum is sensitive to downward mean shifts, this chart can not only give a signal of mean shift but also tell us whether the shift is upward or downward. By tracking the frequency of each component being the maximum or the minimum across all time points, we can further determine the likelihood of each component being shifted. In cases when the process response components have a lower or upper bound, which is common in practice (e.g., the economic indices are always non-negative), the MM-CUSUM chart would be especially efficient. |
