Anisotropic adaptive kernel deconvolution

Abstract

In this paper, we consider a multidimensional convolution model for which we provide adaptive anisotropic kernel estimators of a signal density $f$ measured with additive error. For this, we generalize Fan's~(1991) estimators to multidimensional setting and use a bandwidth selection device in the spirit of Goldenschluger and Lepski's~(2011) proposal fr density estimation without noise. We consider first the pointwise setting and then, we study the integrated risk. Our estimators depend on an automatically selected random bandwidth. We assume both ordinary and super smooth components for measurement errors, which have known density. We also consider both anisotropic H\{o}lder and Sobolev classes for $f$. We provide non asymptotic risk bounds and asymptotic rates for the resulting data driven estimator, which is proved to be adaptive. We provide an illustrative simulation study, involving the use of Fast Fourier Transform algorithms. We conclude by a proposal of extension of the method to the case of unknown noise density, when a preliminary pure noise sample is available.