Deconvolution with unknown error distribution

We consider the problem of estimating a density $f_{X}$ using a sample $Y_{1},...,Y_{n}$ from $f_{Y}=f_{X}*f_{\epsilon}$, where $f_{\epsilon}$ is an unknown density function. We assume that an additional sample $\epsilon_{1},...,\epsilon_{m}$ from $f_{\epsilon}$ is given. Estimators of $f_{X}$ and its derivatives are constructed using nonparametric estimators of $f_{Y}$ and $f_{\epsilon}$ 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 $f_{\epsilon}$ assuming that $f_{X}$ belongs to a Sobolev space $H_{p}$ and that the Fourier transform of $f_{\epsilon}$ 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 $f_{\epsilon}$ 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 $f_{\epsilon}$ compared with the estimator in the case when $f_{\epsilon}$ is known.