Adaptive quantile estimation in deconvolution with unknown error distribution

We study the problem of quantile estimation in deconvolution with ordinary smooth error distributions. In particular, we focus on the more realistic setup of unknown error distributions. We develop a minimax optimal procedure and construct an adaptive estimation method under natural conditions on the densities. As a side result we obtain minimax optimal rates for the plug-in estimation of distribution functions with unknown error distributions. Some numerical results are presented and the application of our estimator to a real data example is studied.