A rank-based multivariate CUSUM procedure

We consider statistical process control when measurements are multivariate. A cumulative sum (CUSUM) procedure is suggested in detecting a shift in the mean vector of the measurements, which is based on the cross-sectional antiranks of the measurements. At each time point, the measurements are ordered and their antiranks, which are the indices of the order statistics, are recorded. When the process is in control and the joint distribution of the multivariate measurements satisfies some regularity conditions, the antirank vector at each time point has a given distribution. This distribution changes to some other distribution when the process is out of control and the components of the shift in the mean vector of the process are not all the same. This CUSUM can therefore detect shifts in all directions except the one in which the components of the shift in the mean vector are all the same but not 0. The shift with equal components, however, can be easily detected by another univariate CUSUM. The former CUSUM procedure is distribution free in the sense that all its properties depend on the distribution of the antirank vector only.