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

TL;DR

This paper establishes a functional central limit theorem for sequential U-quantile processes under dependence, enabling robust change-point detection without bootstrap methods, with applications demonstrated on real data.

Contribution

It introduces a new limit theorem for U-quantile processes under dependence and develops a consistent long-run variance estimator for change-point analysis.

Findings

The studentized U-quantile process converges to a standard Brownian motion.

The proposed change-point test based on U-quantiles is robust and efficient.

Simulation and real data analyses confirm the method's effectiveness.

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 sequential $U$-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 sequential $U$-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 at the example of the Hodges-Lehmann estimator for robustly detecting changes in the central location. A simulation study confirms the very good robustness and efficiency properties of the test. Two real-life data sets are analyzed.