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.

翻译：在变点分析中，U统计量的应用已受到文献的广泛关注。我们比较了两种构建CUSUM型变点检验的方法，分别称为“首对全”与“首对末”方法。这两种方法均已被不同研究者采用。一个自然的问题是：这两种检验是否存在显著差异？如果存在，在何种数据情境下哪一种方法更优？在大样本情况下，两种检验较为相似：在原假设及局部备择序列下，它们是渐近等价的。而在小样本中，可能存在相当明显的差异，这与它们在固定备择下不同的渐近行为一致。我们推导出一个简单的准则，用于判断哪种检验具有更高的功效。我们详细研究了基尼平均差、样本方差和肯德尔τ等实例。特别地，当使用基尼平均差检验尺度变化时，我们证明“首对全”方法具有更高功效的充要条件是尺度从较小值变为较大值——无论总体分布如何或变点位置在何处。渐近推导基于弱相依假设。数值模拟与数据实例验证了上述结果。