Bahadur Representation for U-Quantiles of Dependent Data

TL;DR

This paper extends Bahadur representation results for U-quantiles to dependent data, specifically strongly mixing and absolutely regular sequences, providing new limit theorems and improving existing results for mixing data.

Contribution

It generalizes Bahadur representation for U-quantiles to dependent sequences and derives CLT and LIL results, enhancing understanding of quantiles in dependent data contexts.

Findings

Established Bahadur representation for U-quantiles of dependent data.

Derived central limit theorem and law of the iterated logarithm for these quantiles.

Improved existing results for sample quantiles in mixing data.

Abstract

U-quantiles are applied in robust statistics, like the Hodges-Lehmann estimator of location for example. They have been analyzed in the case of independent random variables with the help of a generalized Bahadur representation. Our main aim is to extend these results to U-quantiles of strongly mixing random variables and functionals of absolutely regular sequences. We obtain the central limit theorem and the law of the iterated logarithm for U-quantiles as straightforward corollaries. Furthermore, we improve the existing result for sample quantiles of mixing data.