  Quantile estimation in deconvolution problems is studied comprehensively. In
particular, the more realistic setup of unknown error distributions is covered.
Our plug-in method is based on a deconvolution density estimator and is minimax
optimal under minimal and natural conditions. This closes an important gap in
the literature. Optimal adaptive estimation is obtained by a data-driven
bandwidth choice. As a side result, we obtain optimal rates for the plug-in
estimation of distribution functions with unknown error distributions. The
method is applied to a real data example.
