DECONVOLUTION WITH UNKNOWN ERROR DISTRIBUTION By Jan Johannes Ruprecht – Karls

We consider the problem of estimating a density fX using a sample Y1,. .. , Yn from fY = fX ⋆ fǫ, where fǫ is an unknown density. We assume that an additional sample ǫ1,. .. , ǫm from fǫ is observed. Estimators of fX and its derivatives are constructed by using non-parametric estimators of fY and fǫ and by applying a spectral cutoff in the Fourier domain. We derive the rate of convergence of the esti-mators in case of a known and unknown error density fǫ, where it is assumed that fX satisfies a polynomial, logarithmic or general source condition. It is shown that the proposed estimators are asymptoti-cally optimal in a minimax sense in the models with known or unknown error density, if the density fX belongs to a Sobolev space HÔ and fǫ is ordinary smooth or supersmooth.