Deconvolution with unknown error distribution

Abstract

We consider the problem of estimating a density f_X using a sample Y₁, …, Y_n from f_Y=f_X⋆f_ε, where f_ε is an unknown density. We assume that an additional sample ε₁, …, ε_m from f_ε is observed. Estimators of f_X and its derivatives are constructed by using nonparametric estimators of f_Y and f_ε 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 f_ε, where it is assumed that f_X 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 f_X belongs to a Sobolev space $H_{\mathcal{p}}$ and f_ε is ordinary smooth or supersmooth.