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.

We analyze the ordinal structure of long‐range dependent time series. To this end, we use so called ordinal patterns which describe the relative position of consecutive data points. We provide two estimators for the probabilities of ordinal patterns and prove limit theorems in different settings, namely stationarity and (less restrictive) stationary increments. In the second setting, we encounter a Rosenblatt distribution in the limit. We prove more general limit theorems for functions with Hermite rank 1 and 2. We derive the limit distribution for an estimation of the Hurst parameter H if it is higher than 3/4. Thus, our theorems complement results for lower values of H which can be found in the literature. Finally, we provide some simulations that illustrate our theoretical results.

We consider an estimator for the location of a shift in the mean of long-range dependent sequences. The estimation is based on the two-sample Wilcoxon statistic. Consistency and the rate of convergence for the estimated change point are established. In the case of a constant shift height, the 1/n convergence rate (with n denoting the number of observations), which is typical under the assumption of independent observations, is also achieved for long memory sequences. It is proved that if the change point height decreases to 0 with a certain rate, the suitably standardized estimator converges in distribution to a functional of a fractional Brownian motion. The estimator is tested on two well-known data sets. Finite sample behaviors are investigated in a Monte Carlo simulation study. Primary 62G05, 62M10; secondary 60G15, 60G22.

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In this article, we show that the recently introduced ordinal pattern dependence fits into the axiomatic framework of general multivariate dependence measures. Furthermore, we consider multivariate generalizations of established univariate dependence measures like Kendall's τ , Spearman's ρ and Pearson's correlation coefficient. Among these, only multivariate Kendall's τ proves to take the dynamical dependence of random vectors stemming from multidimensional time series into account. Consequently, the article focuses on a comparison of ordinal pattern dependence and multivariate Kendall's τ . To this end, limit theorems for multivariate Kendall's τ are established under the assumption of near epoch dependent, data-generating time series. We analyze how ordinal pattern dependence compares to multivariate Kendall's τ and Pearson's correlation coefficient on theoretical grounds. Additionally, a simulation study illustrates differences in the kind of dependencies that are revealed by multivariate Kendall's τ and ordinal pattern dependence.