Deconvolution with unknown error distribution

We consider the problem of estimating a density fX using a sample Y1, . . . , Yn from fY = fX ? fε, where fε is an unknown density function. We assume that an additional sample ε1, . . . , εm from fε is given. Estimators of fX and its derivatives are constructed using nonparametric estimators of fY and fε and applying a spectral cut-off in the Fourier domain. In this paper the rate of convergence of the estimator is derived in the case of a known and an unknown density fε assuming that fX belongs to a Sobolev space Hp and that the Fourier transform of fε descents polynomial, exponential or in some general form. It is shown that the proposed estimator is asymptotically optimal in a minimax sense if the density fε is known or has a Fourier transform with polynomial descent. Monte Carlo simulations demonstrate the reasonable performance of the estimator given an estimated error density fε compared with the estimator in the case when fε is known.