We consider the problem of estimating a density fX using a sample Y1, …, Yn from fY=fX⋆fε, where fε is an unknown density. We assume that an additional sample ε1, …, εm from fε is observed. Estimators of fX and its derivatives are constructed by using nonparametric estimators of fY and fε and by applying a spectral cut-off in the Fourier domain. We derive the rate of convergence of the estimators in case of a known and unknown error density fε, where it is assumed that fX satisfies a polynomial, logarithmic or general source condition. It is shown that the proposed estimators are asymptotically optimal in a minimax sense in the models with known or unknown error density, if the density fX belongs to a Sobolev space $H_{\mathcal{p}}$ and fε is ordinary smooth or supersmooth.
@article{1247663756,
author = {Johannes, Jan},
title = {Deconvolution with unknown error distribution},
journal = {Ann. Statist.},
volume = {37},
number = {1},
year = {2009},
pages = { 2301-2323},
language = {en},
url = {http://dml.mathdoc.fr/item/1247663756}
}
Johannes, Jan. Deconvolution with unknown error distribution. Ann. Statist., Tome 37 (2009) no. 1, pp. 2301-2323. http://gdmltest.u-ga.fr/item/1247663756/