Multivariate quality control using finite intersection tests

Multivariate quality control problems involve the evaluation of a process based on the simultaneous behavior of p variables. Most multivariate quality control procedures evaluate the in-control or out-of-control condition based upon an overall statistic, like Hotelling's T-2. Although T-2 is optimal for finding a general shift in the mean vector, it is not optimal for shifts that occur for some subset of variables, a variable at a time. When this occurs, the optimal procedure is to utilize a Finite Intersection Test (FIT). In this article we show how to use a single step and stepdown FIT to evaluate whether a multivariate process is in control or out of control.