The asymptotic relative efficiency of the mean deviation with respect to the standard deviation is 88% at the normal distribution. In his seminal 1960 paper A survey of sampling from contaminated distributions, J. W. Tukey points out that, if the normal distribution is contaminated by a small -fraction of a normal distribution with three times the standard deviation, the mean deviation is more efficient than the standard deviationalready for < 1%. This came as a surprise to most statisticians at the time, and the publication is today considered as one of the main pioneering works in the development of robust statistics. In the present article, we examine the efficiency of the mean deviation and Gini's mean difference (the mean of all pairwise distances). The latter is known to have an asymptotic relative efficiency of 98% at the normal distribution. Our findings support the viewpoint that Gini's mean difference combines the advantages of the mean deviation and the standard deviation. We also answer the question, what percentage of contamination in Tukey's 1:3 normal mixture model renders Gini's mean difference more efficient than the standard deviation. MSC: 62G35, 62G05, 62G20Date: May 21, 2014.

We introduce a robust estimator of the location parameter for the change-point in the mean based on Wilcoxon statistic and establish its consistency for L 1 near-epoch dependent processes. It is shown that the consistency rate depends on the magnitude of the change. A simulation study is performed to evaluate the finite sample properties of the Wilcoxon-type estimator under Gaussianity as well as under heavy-tailed distributions and disturbances by outliers, and to compare it with a CUSUM-type estimator. It shows that the Wilcoxon-type estimator is equivalent to the CUSUM-type estimator under Gaussianity but outperforms it in the presence of heavy tails or outliers in the data.

In many applications it is important to know whether the amount of fluctuation in a series of observations changes over time. In this article, we investigate different tests for detecting change in the scale of mean-stationary time series. The classical approach based on the CUSUM test applied to the squared centered, is very vulnerable to outliers and impractical for heavy-tailed data, which leads us to contemplate test statistics based on alternative, less outlier-sensitive scale estimators.It turns out that the tests based on Gini's mean difference (the average of all pairwise distances) or generalized Qn estimators (sample quantiles of all pairwise distances) are very suitable candidates. They improve upon the classical test not only under heavy tails or in the presence of outliers, but also under normality. An explanation for this counterintuitive result is that the corresponding long-run variance estimates are less affected by a scale change than in the case of the sample-variance-based test.We use recent results on the process convergence of U-statistics and U-quantiles for dependent sequences to derive the limiting distribution of the test statistics and propose estimators for the long-run variance. We perform a simulations study to investigate the finite sample behavior of the test and their power. Furthermore, we demonstrate the applicability of the new change-point detection methods at two real-life data examples from hydrology and finance.

In this article we introduce a robust to outliers Wilcoxon change-point testing procedure, for distinguishing between short-range dependent time series with a change in mean at unknown time and stationary long-range dependent time series. We establish the asymptotic distribution of the test statistic under the null hypothesis for L 1 near epoch dependent processes and show its consistency under the alternative. The Wilcoxon-type testing procedure similarly as the CUSUM-type testing procedure (of Berkes I., Horváth L., Kokoszka P. and Shao Q. 2006. Ann.Statist. 34:1140-1165), requires estimation of the location of a possible change-point, and then using pre-and post-break subsamples to discriminate between short and long-range dependence. A simulation study examines the empirical size and power of the Wilcoxon-type testing procedure in standard cases and with disturbances by outliers. It shows that in standard cases the Wilcoxon-type testing procedure behaves equally well as the CUSUM-type testing procedure but outperforms it in presence of outliers. We also apply both testing procedure to hydrologic data.

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