Adaptive wavelet multivariate regression with errors in variables

TL;DR

This paper introduces an adaptive wavelet-based method for nonparametric multivariate regression with errors in variables, achieving optimal convergence rates through a data-driven wavelet resolution selection.

Contribution

It develops a novel adaptive estimator using wavelet projection kernels and deconvolution, with a fully automatic procedure for wavelet level selection, and provides theoretical guarantees.

Findings

Achieves optimal convergence rates over anisotropic Hölder classes.

Provides an oracle inequality for the estimator.

Demonstrates effectiveness through simulations.

Abstract

In the multidimensional setting, we consider the errors-in-variables model. We aim at estimating the unknown nonparametric multivariate regression function with errors in the covariates. We devise an adaptive estimator based on projection kernels on wavelets and a deconvolution operator. We propose an automatic and fully data driven procedure to select the wavelet level resolution. We obtain an oracle inequality and optimal rates of convergence over anisotropic H{\"o}lder classes. Our theoretical results are illustrated by some simulations.