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Affiliation(s)
- Herold Dehling
- Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany
| | - Kata Vuk
- Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany
| | - Martin Wendler
- Institut für Mathematische Stochastik, Otto-von-Guericke-Universität Magdeburg, 39106 Magdeburg, Germany
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Affiliation(s)
| | | | - Roland Fried
- Department of Statistics, TU Dortmund University
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Affiliation(s)
- Annika Betken
- Fakultät für Mathematik Ruhr‐Universität Bochum Germany
| | | | | | - Ines Münker
- Department of Mathematics University of Siegen Germany
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Abstract
Summary
We present a robust and nonparametric test for the presence of a changepoint in a time series, based on the two-sample Hodges–Lehmann estimator. We develop new limit theory for a class of statistics based on two-sample U-quantile processes in the case of short-range dependent observations. Using this theory, we derive the asymptotic distribution of our test statistic under the null hypothesis of a constant level. The proposed test shows better overall performance under normal, heavy-tailed and skewed distributions than several other modifications of the popular cumulative sums test based on U-statistics, one-sample U-quantiles or M-estimation. The new theory does not involve moment conditions, so any transform of the observed process can be used to test the stability of higher-order characteristics such as variability, skewness and kurtosis.
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Affiliation(s)
- H Dehling
- Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany
| | - R Fried
- Department of Statistics, TU Dortmund University, 44221 Dortmund, Germany
| | - M Wendler
- Institut für Mathematische Stochastik, Otto von Guericke Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
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Affiliation(s)
- Alexander Schnurr
- Department of Mathematics, University Siegen, Siegen, Germany, and Faculty of Mathematics, Technische Universitat Dortmund, Dortmund, Germany
| | - Herold Dehling
- Department of Mathematics, Ruhr University Bochum, Bochum, Germany
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Affiliation(s)
- H. Dehling
- Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany
| | - A. Rooch
- Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany
| | - M. Wendler
- Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität, Greifswald, Germany
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Abstract
We study the behavior of the empirical distribution function of iterates of intermittent maps in the Hilbert space of square integrable functions with respect to Lebesgue measure. In the long-range dependent case, we prove that the empirical distribution function, suitably normalized, converges to a degenerate stable process, and we give the corresponding almost sure result. We apply the results to the convergence of the Wasserstein distance between the empirical measure and the invariant measure. We also apply it to obtain the asymptotic distribution of the corresponding Cramér–von-Mises statistic.
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Affiliation(s)
- Jérôme Dedecker
- Université Paris Descartes, Sorbonne Paris Cité, Laboratoire MAP5 and CNRS UMR 8145, 75016 Paris, France
| | - Herold Dehling
- Faculty of Mathematics, Ruhr-University Bochum, 44780 Bochum, Germany
| | - Murad S. Taqqu
- Department of Mathematics, Boston University, Boston, MA 02215, USA
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Dehling H, Sharipov OS, Wendler M. Bootstrap for dependent Hilbert space-valued random variables with application to von Mises statistics. J MULTIVARIATE ANAL 2015. [DOI: 10.1016/j.jmva.2014.09.011] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Dehling H, Durieu O, Tusche M. Approximating class approach for empirical processes of dependent sequences indexed by functions. BERNOULLI 2014. [DOI: 10.3150/13-bej525] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Wied D, Dehling H, van Kampen M, Vogel D. A fluctuation test for constant Spearman’s rho with nuisance-free limit distribution. Comput Stat Data Anal 2014. [DOI: 10.1016/j.csda.2013.03.005] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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Dehling H, Durieu O, Tusche M. A sequential empirical CLT for multiple mixing processes with application to $\mathcal{B}$-geometrically ergodic Markov chains. ELECTRON J PROBAB 2014. [DOI: 10.1214/ejp.v19-3216] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Dehling H. André Dabrowski's work on limit theorems and weak dependence. CAN J STAT 2009. [DOI: 10.1002/cjs.10025] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
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Dehling H, Sharipov OSH. Estimation of Mean and Covariance Operator for Banach Space Valued Autoregressive Processes with Dependent Innovations. Stat Infer Stoch Process 2005. [DOI: 10.1007/s11203-003-0382-8] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Dehling H, Steerneman T. Editorial introduction. STAT NEERL 2005. [DOI: 10.1111/j.1467-9574.2005.00300.x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Dehling H, Denker M, Philipp W. A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics. Probab Theory Relat Fields 1986. [DOI: 10.1007/bf00343899] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Dabrowski A, Dehling H, Philipp W. An almost sure invariance principle for triangular arrays of banach space valued random variables. Probab Theory Relat Fields 1984. [DOI: 10.1007/bf00531835] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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