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

Daniel Vogel*, Martin Wendler

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)
10 Downloads (Pure)


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 languageEnglish
Pages (from-to)3114-3144
Number of pages31
Issue number4B
Early online date23 May 2017
Publication statusPublished - 30 Nov 2017

Bibliographical note

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.


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


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