Bayesian quantile estimation in deconvolution

Estimating quantiles of a population is a fundamental problem of high practical relevance in nonparametric statistics. This chapter addresses the problem of quantile estimation in deconvolution models with known error distributions taking a Bayesian approach. We develop the analysis for error distributions with characteristic functions decaying polynomially fast, the so-called ordinary smooth error distributions that lead to mildly ill-posed inverse problems. Using Fourier inversion techniques, we derive an inequality relating the sup-norm distance between mixture densities to the Kolmogorov distance between the corresponding mixing cumulative distribution functions. Exploiting this smoothing inequality, we show that a careful choice of the prior law acting as an efficient approximation scheme for the sampling density leads to adaptive posterior contraction rates to the regularity level of the latent mixing density, thus yielding a new adaptive quantile estimation procedure.