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

TL;DR

This paper derives the asymptotic distribution of delay times for sequential change point tests based on U-statistics, including a new Wilcoxon test, and compares their detection delays for early and late change points.

Contribution

It introduces the asymptotic distribution of delay times for both early and late change points, including a new robust Wilcoxon test, and compares detection delays under different distribution tails.

Findings

Wilcoxon test has smaller delay for heavy-tailed distributions

Asymptotic distribution approximates finite sample delays well

Results extend to late change points, not just early ones

Abstract

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.