Asymptotics of nonparametric L-1 regression models with dependent data

TL;DR

This paper studies the asymptotic behavior of nonparametric L-1 regression estimates with dependent data, providing theoretical insights and uniform Bahadur representations under general dependence structures.

Contribution

It introduces new asymptotic results and Bahadur representations for median quantile estimates in dependent data settings, extending nonparametric regression theory.

Findings

Established uniform Bahadur representations for median quantile estimates.

Analyzed the modulus of continuity of kernel weighted empirical processes.

Illustrated the theoretical results with progesterone data.

Abstract

We investigate asymptotic properties of least-absolute-deviation or median quantile estimates of the location and scale functions in nonparametric regression models with dependent data from multiple subjects. Under a general dependence structure that allows for longitudinal data and some spatially correlated data, we establish uniform Bahadur representations for the proposed median quantile estimates. The obtained Bahadur representations provide deep insights into the asymptotic behavior of the estimates. Our main theoretical development is based on studying the modulus of continuity of kernel weighted empirical process through a coupling argument. Progesterone data is used for an illustration.