DECONVOLUTION WITH UNKNOWN ERROR DISTRIBUTION

We consider the problem of estimating a density f(X) using a sample Y-l,....Y-n from f(Y) = f(X) star f(is an element of), where f(is an element of) is an unknown density. We assume that all additional sample is an element of(l),...,is an element of(m) from f(is an element of) is observed. Estimators of f(X) and its derivatives are constructed by using nonparametric estimators of f(X) and f(is an element of) and by applying a spectral cut-off in the Fourier domain. We derive the rate of convergence of the estimators ill case of a known and unknown error density 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 ill a minimax sense ill the models with known or unknown error density, if the density f(X) belongs to a Sobolev space H-p and f(is an element of) is ordinary smooth or supersmooth.