## Bernoulli

• Bernoulli
• Volume 23, Number 4B (2017), 3114-3144.

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

#### Abstract

Many popular robust estimators are $U$-quantiles, most notably the Hodges–Lehmann location estimator and the $Q_{n}$ 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.

#### Article information

Source
Bernoulli, Volume 23, Number 4B (2017), 3114-3144.

Dates
Revised: March 2016
First available in Project Euclid: 23 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1495505086

Digital Object Identifier
doi:10.3150/16-BEJ838

Mathematical Reviews number (MathSciNet)
MR3654800

Zentralblatt MATH identifier
06778280

#### Citation

Vogel, Daniel; Wendler, Martin. Studentized U -quantile processes under dependence with applications to change-point analysis. Bernoulli 23 (2017), no. 4B, 3114--3144. doi:10.3150/16-BEJ838. https://projecteuclid.org/euclid.bj/1495505086

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