Studentized U-quantile processes under dependence with applications to change-point analysis

Daniel Vogel; Martin Wendler

doi:10.3150/16-BEJ838

Studentized U-quantile processes under dependence with applications to change-point analysis

Daniel Vogel^*
, Martin Wendler

^*Corresponding author for this work

University of Greifswald

Research output: Contribution to journal › Article › peer-review

10 Citations (Scopus)

24 Downloads (Pure)

Abstract

Many popular robust estimators are U-quantiles, most notably the Hodges–Lehmann location estimator and the Qn scale estimator. We prove a functional central limit theorem for the 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 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 with the example of the Hodges–Lehmann estimator for robustly detecting changes in the central location. A simulation study confirms the very good efficiency and robustness properties of the test. Two real-life data sets are analyzed.

Original language	English
Pages (from-to)	3114-3144
Number of pages	31
Journal	Bernoulli
Volume	23
Issue number	4B
Early online date	23 May 2017
DOIs	https://doi.org/10.3150/16-BEJ838
Publication status	Published - 30 Nov 2017

Bibliographical note

Acknowledgement
The research was supported by the DFG Sonderforschungsbereich 823 (Collaborative Research Center) Statistik nichtlinearer dynamischer Prozesse. The authors thank Svenja Fischer and Wei Biao Wu for providing the river Elbe discharge data set and the Argentina rainfall data set, respectively. We are very grateful to the anonymous referee for the comments, which have helped to improve and clarify this manuscript.

Keywords

CUSUM test
Hodges–Lehmann estimator
Long-run variance
Median
Near epoch dependence
Robustness
Weak invariance principle.

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10.3150/16-BEJ838Licence: Unspecified

Studentized U-quantile processes under dependence with applications to change-point analysis
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Cite this

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title = "Studentized U-quantile processes under dependence with applications to change-point analysis",

abstract = "Many popular robust estimators are U-quantiles, most notably the Hodges{\textendash}Lehmann location estimator and the Qn scale estimator. We prove a functional central limit theorem for the 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 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 with the example of the Hodges{\textendash}Lehmann estimator for robustly detecting changes in the central location. A simulation study confirms the very good efficiency and robustness properties of the test. Two real-life data sets are analyzed.",

keywords = "CUSUM test, Hodges{\textendash}Lehmann estimator, Long-run variance, Median, Near epoch dependence, Robustness, Weak invariance principle.",

author = "Daniel Vogel and Martin Wendler",

note = "Acknowledgement The research was supported by the DFG Sonderforschungsbereich 823 (Collaborative Research Center) Statistik nichtlinearer dynamischer Prozesse. The authors thank Svenja Fischer and Wei Biao Wu for providing the river Elbe discharge data set and the Argentina rainfall data set, respectively. We are very grateful to the anonymous referee for the comments, which have helped to improve and clarify this manuscript.",

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doi = "10.3150/16-BEJ838",

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AU - Wendler, Martin

N1 - Acknowledgement The research was supported by the DFG Sonderforschungsbereich 823 (Collaborative Research Center) Statistik nichtlinearer dynamischer Prozesse. The authors thank Svenja Fischer and Wei Biao Wu for providing the river Elbe discharge data set and the Argentina rainfall data set, respectively. We are very grateful to the anonymous referee for the comments, which have helped to improve and clarify this manuscript.

PY - 2017/11/30

Y1 - 2017/11/30

N2 - Many popular robust estimators are U-quantiles, most notably the Hodges–Lehmann location estimator and the Qn scale estimator. We prove a functional central limit theorem for the 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 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 with the example of the Hodges–Lehmann estimator for robustly detecting changes in the central location. A simulation study confirms the very good efficiency and robustness properties of the test. Two real-life data sets are analyzed.

AB - Many popular robust estimators are U-quantiles, most notably the Hodges–Lehmann location estimator and the Qn scale estimator. We prove a functional central limit theorem for the 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 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 with the example of the Hodges–Lehmann estimator for robustly detecting changes in the central location. A simulation study confirms the very good efficiency and robustness properties of the test. Two real-life data sets are analyzed.

KW - CUSUM test

KW - Hodges–Lehmann estimator

KW - Long-run variance

KW - Median

KW - Near epoch dependence

KW - Robustness

KW - Weak invariance principle.

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U2 - 10.3150/16-BEJ838

DO - 10.3150/16-BEJ838

M3 - Article

SN - 1573-9759

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EP - 3144

JO - Bernoulli

JF - Bernoulli

IS - 4B

ER -

Studentized U-quantile processes under dependence with applications to change-point analysis

Abstract

Bibliographical note

Keywords

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