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

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.

💡 Research Summary

The paper investigates two CUSUM‑type change‑point tests built from U‑statistics, referred to as the “first‑vs‑full” and the “first‑vs‑last” approaches. Both methods have appeared in the literature, but it has not been clear whether they differ substantially in practice and, if so, under which circumstances one should be preferred. The authors first establish that under the null hypothesis of no change, the two test statistics are asymptotically equivalent: they converge to the same Gaussian limit after appropriate standardisation, even when the underlying observations exhibit weak dependence (α‑mixing). This result also holds for sequences of local alternatives, i.e., alternatives in which the magnitude of the change shrinks at the n‑½ rate. Consequently, for large samples the two procedures have essentially identical size and power.

The core contribution lies in the analysis of fixed alternatives, where the change magnitude does not vanish with the sample size. In this regime the two statistics have different deterministic limits, leading to distinct asymptotic power functions. The authors derive a simple analytical criterion that tells which test is more powerful for a given alternative. The criterion compares the signed difference between the pre‑change and post‑change values of the underlying parameter and the way this difference is weighted in each statistic.

To illustrate the theory, three widely used U‑statistics are examined in detail:

1. Gini’s mean difference (GMD) – the average absolute difference between all pairs of observations, which is a scale‑sensitive measure.
2. Sample variance – the second‑order central moment, also a scale measure but with different weighting.
3. Kendall’s τ – a rank‑based correlation coefficient, sensitive to changes in dependence structure.

For each case the authors compute the asymptotic power expressions for both tests and apply the derived criterion. A striking result emerges for GMD: the first‑vs‑full test is uniformly more powerful than the first‑vs‑last test iff the scale parameter increases (i.e., the distribution becomes more dispersed after the change). This dominance holds regardless of the underlying distribution (normal, heavy‑tailed, skewed, etc.) and irrespective of the change point’s location within the sample. Conversely, when the scale decreases, the first‑vs‑last test dominates. For the sample variance the same pattern is observed, while for Kendall’s τ the direction of superiority depends on whether the rank correlation strengthens or weakens.

The theoretical findings are corroborated by extensive Monte‑Carlo simulations. The simulations cover independent data as well as weakly dependent time series generated from AR(1), MA(1), ARMA(1,1), and GARCH(1,1) models. Results confirm that the asymptotic equivalence under the null and local alternatives is accurate even for moderate sample sizes, and that the power differences predicted under fixed alternatives become pronounced when the sample size is small to moderate (e.g., n ≈ 50–200).

Two real‑data applications demonstrate practical relevance. In a financial return series, the first‑vs‑full test based on GMD detects a volatility increase earlier and with higher significance than the first‑vs‑last test. In an environmental monitoring series of air‑pollutant concentrations, where the dependence structure weakens, the first‑vs‑last test based on Kendall’s τ identifies the change more effectively. These examples illustrate that the choice between the two procedures should be guided by the nature of the anticipated change (scale up vs. scale down, variance vs. dependence) and by sample‑size considerations.

In summary, the paper provides a comprehensive theoretical comparison of two popular U‑statistic change‑point tests, establishes their asymptotic equivalence under the null and local alternatives, and supplies a clear, analytically tractable rule for selecting the more powerful test under fixed alternatives. The results are derived under weak dependence, making them applicable to a broad class of time‑series data. Practitioners can now decide, before analysis, whether to employ the first‑vs‑full or the first‑vs‑last approach based on the expected direction of change and the specific U‑statistic of interest, thereby improving detection power in finite‑sample settings.