Deconvolution for some singular density errors via a combinatorial median of means approach

We present a versatile and model based procedure for estimating a density in a deconvolution setting where the error density is assumed to be singular enough.We assess the quality of our estimator by establishing non-asymptotic risk bounds for the L^1 loss. We specify them when the density is piecewise constant on a finite number of (unknown) pieces, when it is unimodal, and when it is concave/convex.