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

April 3rd, 2014

A new preprint joint with Aeneas Rooch 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.

New Preprint: Stable Limit Theorem for U-Statistic Processes Indexed by a Random Walk

Dezember 12th, 2012

A new preprint joint with Brice Franke about “Stable Limit Theorem for U-Statistic Processes Indexed by a Random Walk” is online at arXiv.

Abstract: Let (S_n)_{n\in\N} be a random walk in the domain of attraction of an α-stable Lévy process and (\xi(n))_{n\in\N} a sequence of iid random variables (called scenery). We want to investigate U-statistics indexed by the random walk S_n, that is U_n:=\sum_{1\leq i< j\leq n}h(\xi(S_i),\xi(S_j)) for some symmetric bivariate function h. We will prove the weak convergence without the assumption of finite variance. Additionally, under the assumption of finite moments of order greater than two, we will establish a law of the iterated logarithm for the U-statistic U_n.