Bayesian quantile estimation in deconvolution

Estimating quantiles of a population is a fundamental problem in nonparametric statistics, with high practical relevance. This note deals, from a Bayesian point of view, with quantile estimation in deconvolution problems with known error distribution. We pursue the analysis for error distributions whose characteristic functions decay polynomially fast, the so-called ordinary smooth errors that lead to mildly ill-posed inverse problems. Our method is based on Fourier inversion techniques for density deconvolution and the estimation procedure for single quantiles is minimax-optimal under minimal conditions.