Asymptotic distribution of two-sample empirical U-quantiles with applications to robust tests for structural change

Abstract

We derive the asymptotical distributions of two-sample U-statistics and two-sample empirical U-quantiles in the case of weakly dependent data. Our results apply to observations that can be represented as functionals of absolutely regular processes, including e.g. many classical time series models as well as data from chaotic dynamical systems. Based on these theoretical results we propose a new robust nonparametric test for the two-sample location problem, which is constructed from the median of pairwise differences between the two samples. We inspect the properties of the test in the case of weakly dependent data and compare the performance with classical tests such as the t-test and Wilcoxon’s two-sample rank test with corrections for dependencies. Simulations indicate that the new test offers better power even than the Wilcoxon test in case of skewed and heavy tailed distributions, if at least one of the two samples is not very large. The test is then applied for detecting shifts of location in some weakly dependent time series, which are contaminated by outliers.