First versus full or first versus last: U-statistic change-point tests under fixed and local alternatives

TL;DR

This paper compares two U-statistic based change-point tests, analyzing their performance under different sample sizes and change scenarios, and provides criteria for choosing the more powerful test in various situations.

Contribution

It introduces a simple criterion to determine which of the two U-statistic based change-point tests is more powerful depending on the data situation and change type.

Findings

Both tests are asymptotically equivalent under the null hypothesis.

In small samples, the tests may differ significantly in performance.

The first-vs-full approach is more powerful for detecting increases in scale from smaller to larger values.

Abstract

The use of U-statistics in the change-point context has received considerable attention in the literature. We compare two approaches of constructing CUSUM-type change-point tests, which we call the first-vs-full and first-vs-last approach. Both have been pursued by different authors. The question naturally arises if the two tests substantially differ and, if so, which of them is better in which data situation. In large samples, both tests are similar: they are asymptotically equivalent under the null hypothesis and under sequences of local alternatives. In small samples, there may be quite noticeable differences, which is in line with a different asymptotic behavior under fixed alternatives. We derive a simple criterion for deciding which test is more powerful. We examine the examples Gini's mean difference, the sample variance, and Kendall's tau in detail. Particularly, when testing for changes in scale by Gini's mean difference, we show that the first-vs-full approach has a higher power if and only if the scale changes from a smaller to a larger value -- regardless of the population distribution or the location of the change. The asymptotic derivations are under weak dependence. The results are illustrated by numerical simulations and data examples.