Adaptive estimation in regression models for weakly dependent data and explanatory variable with known density

TL;DR

This paper proposes an adaptive kernel-based estimation method for regression functions with weakly dependent explanatory variables, demonstrating oracle inequalities and adaptivity over Hölder classes, supported by simulations.

Contribution

It introduces a data-driven bandwidth selection procedure for regression estimation with weakly dependent data, ensuring adaptivity and theoretical guarantees.

Findings

Estimator satisfies oracle-type inequality

Method is adaptive over Hölder classes

Simulations illustrate good performance

Abstract

This article is dedicated to the estimation of the regression function when the explanatory variable is a weakly dependent process whose correlation coefficient exhibits exponential decay and has a known bounded density function. The accuracy of the estimation is measured using pointwise risk. A data-driven procedure is proposed using kernel estimation with bandwidth selected via the Goldenshluger-Lepski approach. We demonstrate that the resulting estimator satisfies an oracle-type inequality and it is also shown to be adaptive over H\"older classes. Additionally, unsupervised statistical learning techniques are described and applied to calibrate the method, and some simulations are provided to illustrate the performance of the method.