Abstract. We propose a nonparametric change-point test for long-range dependent data, which is based on the Wilcoxon two-sample test. We derive the asymptotic distribution of the test statistic under the null-hypothesis that no change occured. In a simulation study, we compare the power of our test with the power of a test which is based on differences of means. The results of the simulation study show that in the case of Gaussian data, our test has only slightly smaller power than the difference-of-means test. For heavy-tailed data, our test outperforms the difference-of-means test.

We investigate the power of the CUSUM test and the Wilcoxon change-point tests for a shift in the mean of a process with long-range dependent noise. We derive analytic formulas for the power of these tests under local alternatives. These results enable us to calculate the asymptotic relative efficiency (ARE) of the CUSUM test and the Wilcoxon change point test. We obtain the surprising result that for Gaussian data, the ARE of these two tests equals 1, in contrast to the case of i.i.d. noise when the ARE is known to be 3/π.Key words and phrases. Change-point problems, nonparametric change-point tests, Wilcoxon two-sample rank test, power of test, local alternatives, asymptotic relative efficiency of tests, long-range dependent data, long memory, functional limit theorem.

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Motivated by some common change-point tests, we investigate the asymptotic distribution of the U-statistic processwhen the underlying data are long-range dependent. We present two approaches, one based on an expansion of the kernel h(x, y) into Hermite polynomials, the other based on an empirical process representation of the U-statistic. Together, the two approaches cover a wide range of kernels, including all kernels commonly used in applications. k i=1 n j=k+1 h(X i , X j ). This holds, for example, for the CUSUM test and the Wilcoxon change-point test, where h(x, y) = y − x and h(x, y) = 1 {x≤y} , respectively. The asymptotic distribution of such test statistics can be obtained if one knows the limit distribution of the corresponding two-sample U-statistic process. The asymptotic distribution of the two-sample U-statistic process has been obtained earlier by Horváth (1988), for i.i.d. data, andby Dehling, Fried, Garcia andWendler (2013) for short-range dependent data. In both cases, n −3/2 (U n (t)) 0≤t≤1 converges in distribution, on the space D[0, 1], towards a Gaussian process. In the case of long-range dependent data, the two-sample U-statistic process has been studied only for two specific examples,