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 estimators 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 Hölder classes. Our theoretical results are illustrated by some simulations.
Highlights
Deconvolution problems arise in many fields where data are obtained with measurement errors and has attracted a lot of attention in the statistical literature, see [21] for an excellent source of references
In the one dimensional setting, [3] used Meyer wavelets in order to devise his statistical procedure but his assumptions on the model are strong since the corrupted observations Wl follow a uniform density on [0, 1]. [5] investigated the mean integrated squared error with a penalized estimator based on projection methods upon Shannon basis
Our approach relies on the use of projection kernels on wavelets bases combined with a deconvolution operator involving the noise in the covariates
Summary
For estimating the regression function m, the idea consists in writing m as the ratio m(x) = m(x)fX (x) , x ∈ [0, 1]d. Since estimating fX is a classical deconvolution problem, the main task consists in estimating p. We propose a wavelet-based procedure with an automatic choice of the maximal resolution level.
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