  Sequential change point tests aim at giving an alarm as soon as possible
after a structural break occurs while controlling the asymptotic false alarm
error. For such tests it is of particular importance to understand how quickly
a break is detected. While this is often assessed by simulations only, in this
paper, we derive the asymptotic distribution of the delay time 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. Similar to
asymptotic relative efficiency in an a-posteriori setting, the results allow us
to compare the detection delay of the two procedures. It is shown that the
Wilcoxon sequential procedure has a smaller detection delay for heavier tailed
distributions which is also confirmed by simulations. While the previous
literature only derives results for early change points, we obtain the
asymptotic distribution of the delay time 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.
