Asymptotic delay times of sequential tests based on [formula omitted]-statistics for early and late change points

Abstract

Sequential change point tests aim at giving an alarm as soon as possible after a structural break has occurred while controlling the asymptotic false alarm rate. For such tests it is of particular importance to understand how quickly a break is detected. In this paper, we derive the asymptotic distribution of the delay times for sequential change point procedures based on U-statistics. This includes the difference-of-means (DOM) sequential test that has been discussed previously but also a new robust Wilcoxon sequential change point test. Based on these results we can compare the detection delay of both procedures instead of having to rely on simulations only. Indeed, the Wilcoxon sequential procedure has a smaller detection delay for heavier tailed distributions which is also confirmed by simulations. While in the previous literature for the DOM test only results for early change points were obtained, we derive the asymptotic distribution of the delay times for both early as well as late change points. Finally, we evaluate how well the asymptotic distribution approximates the actual stopping times for finite samples via a simulation study.