Testing for Change‐Points in Long‐Range Dependent Time Series by Means of a Self‐Normalized Wilcoxon Test

Abstract

We propose a testing procedure based on the Wilcoxon two‐sample test statistic in order to test for change‐points in the mean of long‐range dependent data. We show that the corresponding self‐normalized test statistic converges in distribution to a non‐degenerate limit under the hypothesis that no change occurred and that it diverges to infinity under the alternative of a change‐point with constant height. Furthermore, we derive the asymptotic distribution of the self‐normalized Wilcoxon test statistic under local alternatives, that is, under the assumption that the height of the level shift decreases as the sample size increases. Regarding the finite sample performance, simulation results confirm that the self‐normalized Wilcoxon test yields a consistent discrimination between hypothesis and alternative and that its empirical size is already close to the significance level for moderate sample sizes.