Power of Change-Point Tests for Long-Range Dependent Data

We investigate the power of the CUSUM test and the Wilcoxon change-point test for a shift in the mean of a process with long-range dependent noise. We derive analytiv 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 $3/\pi$

Two-Sample U-Statistic Processes for Long-Range Dependent Data

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

Power of change-point tests for long-range dependent data

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.Herold Dehling and Aeneas Rooch were supported in part by the German Research Foundation (DFG) through the Collaborative Research Center SFB 823 Statistical Modelling of Nonlinear Dynamic Processes. Murad S. Taqqu was supported in part by NSF grant DMS-1309009 at Boston University. (SFB 823 - German Research Foundation (DFG); DMS-1309009 - NSF at Boston University)Published versio

Estimation methods for the LRD parameter under a change in the mean

When analyzing time series which are supposed to exhibit long-range dependence (LRD), a basic issue is the estimation of the LRD parameter, for example the Hurst parameter H 2 (1=2; 1). Conventional estimators of H easily lead to spurious detection of long memory if the time series includes a shift in the mean. This defect has fatal consequences in change-point problems: Tests for a level shift rely on H, which needs to be estimated before, but this estimation is distorted by the level shift. We investigate two blocks approaches to adapt estimators of H to the case that the time series includes a jump and compare them with other natural techniques as well as with estimators based on the trimming idea via simulations. These techniques improve the estimation of H if there is indeed a change in the mean. In the absence of such a change, the methods little affect the usual estimation. As adaption, we recommend an overlapping blocks approach: If one uses a consistent estimator, the adaption will preserve this property and it performs well in simulations

Nonparametric change-point tests for long-range dependent data

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

Ein Kreis mit unendlich vielen Mittelpunkten : die erstaunliche Welt der p-adischen Geometrie

Die Welt, die Annette Werner untersucht, erscheint uns fremd, fast schon absurd: Verschiedene Zahlen haben hier die gleiche Größe, und Kreise besitzen unendlich viele Mittelpunkte. Die Mathematikprofessorin forscht auf dem Gebiet der sogenannten p-adischen Geometrie – einem Bereich der modernen Algebra, der in den letzten Jahrzehnten einen stürmischen Fortschritt erlebt hat

Strukturbruchtests für stark abhängige Daten

In der Arbeit "Change-Point Tests For Long-Range Dependent Data" (Dissertation von Aeneas Rooch, Ruhr-Universität Bochum, 2012) werden mathematische Methoden entwickelt, um in Messreihen (wie Börsenkursen oder Temperaturmessungen) zwischen zufälligen Schwankungen und grundlegenden Änderungen in der Natur der Daten, sogenannten Strukturbrüchen, zu unterscheiden. Solche Unterscheidungen sind besonders schwierig bei stark abhängigen Daten, bei denen sich jede einzelne Beobachtung auch noch über große Zeitspannen auf die folgenden Beobachtungen auswirkt. Von Klimamessungen bis zu Datenverkehr in Netzwerken lassen sich viele Prozesse durch stark abhängige Zeitreihen modellieren. Gewöhnliche Verfahren zur Strukturbrucherkennung versagen jedoch bei einer solchen Art von Daten. In der Arbeit werden nicht-parametrische Tests zur Strukturbrucherkennung bei stark abhängigen Daten entwickelt, mathematisch hergeleitet und sowohl analytisch als auch in Simulationsstudien miteinander verglichen.In the thesis "Change-Point Tests For Long-Range Dependent Data" (by Aeneas Rooch, Ruhr-Universität Bochum, 2012), mathematical methods for data analysis are developed. In time series (like stock exchange prices or temperature measurements), it is important to discriminate between random fluctuations and an underlying change in the structure, so called change-points. This is difficult specifically in so called long-range dependent data where even events from the distant past influence the present behaviour. Many important processes like internet traffic and temperature measurements can be modelled by such kind of random time series, but common change-point tests fail when the data is long-range dependent. In this work, limit theorems for non-parametric change-point tests for long-range dependent data are proved. Moreover, some special change-point tests are compared analytically and in large simulation studies