Stopping times for sequential U-Statistics

To understand the speed of detection is of particular importance in sequential change point analysis as, for example, monitoring patient or machine data requires a quick intervention as soon as possible after a structural break has occurred. Therefore, we derive the limit distribution of the delay time for a general framework of sequential change point procedures based on U-statistics for early as well as late change points. The asymptotic delay time for late changes has not been considered in the literature before, not even for the classical sequential CUSUM procedure, and requires different asymptotic considerations. Based on the asymptotic behavior of the delay time we provide a theoretical comparison of the sequential CUSUM procedure and a more robust Wilcoxon-type procedure in terms of their speed of detection. The approximation of the stopping time for finite samples via the limit distribution is evaluated by some simulations.