## New preprint: Studentized sequential U-quantiles under dependence with applications to change-point analysis

März 30th, 2015

A new preprint joint with Daniel Vogel about “Studentized sequential U-quantiles under dependence with applications to change-point Analysis” is online at arXiv.

Abstract: Many popular robust estimators are U-quantiles, most notably the Hodges-Lehmann location estimator and the Q_n scale estimator. We prove a functional central limit theorem for the sequential U-quantile process without any moment assumptions and under weak short-range dependence conditions. We further devise an estimator for the long-run variance and show its consistency, from which the convergence of the studentized version of the sequential U-quantile process to a standard Brownian motion follows. This result can be used to construct CUSUM-type change-point tests based on U-quantiles, which do not rely on bootstrapping procedures. We demonstrate this approach in detail at the example of the Hodges-Lehmann estimator for robustly detecting changes in the central location. A simulation study confirms the very good robustness and efficiency properties of the test. Two real-life data sets are analyzed.

## New preprint: Sequential block bootstrap in a Hilbert space with application to change point analysis

Dezember 20th, 2014

A new preprint joint with Olimjon Sh. Sharipov and Johannes Tewes about “Sequential block bootstrap in a Hilbert space with application to change point analysis” is online at arXiv.

Abstract: A new test for structural changes in functional data is investigated. It is based on Hilbert space theory and critical values are deduced from bootstrap iterations. Thus a new functional central limit theorem for the block bootstrap in a Hilbert space is required. The test can also be used to detect changes in the marginal distribution of random vectors, which is supplemented by a simulation study. Our methods are applied to hydrological data from Germany.

## New preprint: Multivariate generalized linear-statistics of short range dependent data

Dezember 20th, 2014

A new preprint joint with Svenja Fischer and Roland Fried about “Multivariate generalized linear-statistics of short range dependent data” is online at arXiv.

Abstract: Generalized linear (GL-) statistics are defined as functionals of an U-quantile process and unify different classes of statistics such as U-statistics and L-statistics. We derive a central limit theorem for GL-statistics of strongly mixing sequences and arbitrary dimension of the underlying kernel. For this purpose we establish a limit theorem for U-statistics and an invariance principle for U-processes together with a convergence rate for the remaining term of the Bahadur representation. An application is given by the generalized median estimator for the tail-parameter of the Pareto distribution, which is commonly used to model exceedances of high thresholds. We use subsampling to calculate confidence intervals and investigate its behaviour under independence and strong mixing in simulations.

## New preprint: The sequential empirical process of a random walk in random scenery

Oktober 12th, 2014

A new preprint about “The sequential empirical process of a random walk in random scenery” is online at arXiv.

Abstract: A random walk in random scenery $(Y_n)_{n\in\N}$ is given by $Y_n=\xi_{S_n}$ for a random walk $(S_n)_{n\in\N}$ and iid random variables $(\xi(n))_{n\in\N}$. In this paper, we will show the weak convergence of the sequential empirical process, i.e. the centered and rescaled empirical distribution function. The limit process shows a new type of behavior, combining properties of the limit in form independent case (roughness of the paths) and of the long range dependent case (self-similarity).

## New preprint: Simplified simplicial depth for regression and autoregressive growth processes

Oktober 12th, 2014

A new preprint joint with Christoph P. Kustosz and  Christine H. Müller about “Simplified simplicial depth for regression and autoregressive growth processes” is online as a SFB 823 discussion paper.

Abstract: We simplify simplicial depth for regression and autoregressive growth processes in two directions. At first we show that often simplicial depth reduces to counting the subsets with alternating signs of the residuals. The second simplification is given by not regarding all subsets of residuals. By consideration of only special subsets of residuals, the asymptotic distributions of the simplified simplicial depth notions are normal distributions so that tests and confidence intervals can be derived easily. We propose two simplifications for the general case and a third simplification for the special case where two parameters are unknown. Additionally, we derive conditions for the consistency of the tests. We show that the simplified depth notions can be used for polynomial regression, for several nonlinear regression models, and for several autoregressive growth processes. We compare the efficiency and robustness of the different simplified versions by a simulation study concerning the Michaelis-Menten model and a nonlinear autoregressive process of order one.

## Video: Bootstrap for dependent Hilbert space-valued random variables

Mai 17th, 2014

My talk at the conference on “Recent Advances and Trends in Time Series Analysis: Nonlinear Time Series, High Dimensional Inference and Beyond” at the Banff International Research Station was recorded. If you are interested, please follow this link.

## New preprint: Two-sample U-statistic processes for long-range dependent data

April 3rd, 2014

A new preprint joint with Aeneas Rooch and Herold Dehling about “Two-sample U-statistic processes for long-range dependent data” is online at arXiv.

Abstract: Motivated by some common-change point tests, we investigate the asymptotic distribution of the U-statistic process $U_n(t)=\sum_{i=1}^{[nt]}\sum_{j=[nt]+1}^n h(X_i,X_j)$, 0≤t≤1, when 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.

## New preprint: Convergence of U-statistics indexed by a random walk to stochastic integrals of a Levy sheet

Februar 8th, 2014

A new preprint joint with Brice Franke and Françoise Pène about “Convergence of U-statistics indexed by a random walk to stochastic integrals of a Levy sheet” is online at arXiv.

Abstract: We establish limit theorems for U-statistics indexed by a random walk on Z^d and we express the limit in terms of some Levy sheet Z(s,t). Under some hypotheses, we prove that the limit process is Z(t,t) if the random walk is transient or null-recurrent ant that it is some stochastic integral with respect to Z when the walk is positive recurrent. We compare our results with results for random walks in random scenery.

## New Preprint: Bootstrap for dependent Hilbert space-valued random variables with application to von Mises statistics

Dezember 17th, 2013

A new preprint joint with Herold Dehling and Olimjon Sh. Sharipov about “Bootstrap for dependent Hilbert space-valued random variables with application to von Mises statistics” is online at arXiv.

Abstract: Statistical methods for functional data are of interest for many applications. In this paper, we prove a central limit theorem for random variables taking their values in a Hilbert space. The random variables are assumed to be weakly dependent in the sense of near epoch dependence, where the underlying process fulfills some mixing conditions. As parametric inference in an infinite dimensional space is difficult, we show that the nonoverlapping block bootstrap is consistent. Furthermore, we show how these results can be used for degenerate von Mises-statistics.

## New Preprint: Change-Point Detection under Dependence Based on Two-Sample U-Statistics

April 10th, 2013

A new preprint joint with Herold Dehling, Roland Fried and Isabel García about “Change-Point Detection under Dependence Based on Two-Sample U-Statistics” is online at arXiv.

Abstract: We study the detection of change-points in time series. The classical CUSUM statistic for detection of jumps in the mean is known to be sensitive to outliers. We thus propose a robust test based on the Wilcoxon two-sample test statistic. The asymptotic distribution of this test can be derived from a functional central limit theorem for two-sample U-statistics. We extend a theorem of Csorgo and Horvath to the case of dependent data.