Uniform Bahadur Representation for Local Polynomial Estimates of M-Regression and Its Application to The Additive Model

TL;DR

This paper establishes a uniform Bahadur representation for local polynomial M-regression estimators in dependent data, providing a foundation for statistical inference and applications like additive models.

Contribution

It introduces a strong uniform consistency rate for the Bahadur representation of local polynomial M-regression estimators under dependence.

Findings

Established uniform consistency rate for Bahadur representation

Applied results to additive M-regression model estimation

Provided theoretical foundation for inference with dependent data

Abstract

We use local polynomial fitting to estimate the nonparametric M-regression function for strongly mixing stationary processes $\{(Y_{i},\underline{X}_{i})\}$. We establish a strong uniform consistency rate for the Bahadur representation of estimators of the regression function and its derivatives. These results are fundamental for statistical inference and for applications that involve plugging in such estimators into other functionals where some control over higher order terms are required. We apply our results to the estimation of an additive M-regression model.