Adaptive quantile estimation in deconvolution with unknown error distribution

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.