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Sequential change point tests aim at giving an alarm as soon as possibleafter a structural break occurs while controlling the asymptotic false alarmerror. For such tests it is of particular importance to understand how quicklya break is detected. While this is often assessed by simulations only, in thispaper, we derive the asymptotic distribution of the delay time for sequentialchange point procedures based on U-statistics. This includes thedifference-of-means (DOM) sequential test, that has been discussed previously,but also a new robust Wilcoxon sequential change point test. Similar toasymptotic relative efficiency in an a-posteriori setting, the results allow usto compare the detection delay of the two procedures. It is shown that theWilcoxon sequential procedure has a smaller detection delay for heavier taileddistributions which is also confirmed by simulations. While the previousliterature only derives results for early change points, we obtain theasymptotic distribution of the delay time for both early as well as late changepoints. Finally, we evaluate how well the asymptotic distribution approximatesthe actual stopping times for finite samples via a simulation study.

Asymptotic delay times of sequential tests based on U-statistics for early and late change points