Kernel Selection in Nonparametric Regression

TL;DR

This paper develops an oracle inequality for a kernel-based estimator in nonparametric regression, addressing kernel and dimension selection, and extends previous work to anisotropic projection estimators.

Contribution

It introduces a new oracle inequality for kernel and dimension selection in nonparametric regression, including anisotropic projection estimators, using the PCO method.

Findings

Oracle inequality established for kernel estimator with PCO method

Dimension selection for anisotropic projection estimators

Extension of bandwidth selection results to more complex estimators

Abstract

In the regression model $Y = b(X) +\sigma(X)\varepsilon$, where $X$ has a density $f$, this paper deals with an oracle inequality for an estimator of $bf$, involving a kernel in the sense of Lerasle et al. (2016), selected via the PCO method. In addition to the bandwidth selection for kernel-based estimators already studied in Lacour, Massart and Rivoirard (2017) and Comte and Marie (2020), the dimension selection for anisotropic projection estimators of $f$ and $bf$ is covered.