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

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.

Figures (8)

• Figure 1: Visualization of Theorem \ref{['th:fixed']} for Gini's mean difference: A change occurs at $\tau^\star = 1/3$ from $F = N(0,1)$ to $G = N(0,4)$. The limit functions $\Psi_1$ (first-vs-full test; dark blue) and $\Psi_2$ (first-vs-last test; light blue) as given in Theorem \ref{['th:fixed']} along with corresponding trajectories of $\left(\frac{1}{n}D_{n}^F(k)\right)_{1\le k \le n}$ and $\left(\frac{1}{n}D_{n}^L(k)\right)_{1\le k \le n}$ for a sequence of $n$ independent observations with $n = 100$ (left panel) and $n = 4000$ (right panel).
• Figure 2: The plot of Example \ref{['ex:kendall:normal']}: It shows the eccentricity $\rho_{FG}$ on the z-axis as a function of $\rho_1$ and $\rho_2$, i.e., the correlation of the bivariate centered Gaussian time series before and after the change, respectively. In the blue region, the first-vs-full test is more efficient.
• Figure 3: Annual mean discharge ($m^3/s$) of the Colorado river at Lees Ferry, AZ (darkblue) and Cameo, CO (lightblue) from 1934 to 2021.
• Figure 4: Gini's mean difference: studentized change-point processes first-vs-full and first-vs-last applied to the Lees Ferry series.
• Figure 5: Gini's mean difference: studentized change-point processes first-vs-full and first-vs-last applied to the Cameo series.

Theorems & Definitions (47)

• Theorem 2.2
• Remark 2.3
• Theorem 2.6
• Proposition 2.7
• Theorem 2.9
• Lemma 2.10
• Corollary 2.11
• Proposition 2.12
• Theorem 2.13
• Proposition 3.1