Adaptivity in convolution models with partially known noise distribution

Abstract: We consider a semiparametric convolution model. We observe random variables having a distribution given by the convolution of some unknown density f and some partially known noise density g. In this work, g is assumed exponentially smooth with stable law having unknown selfsimilarity index s. In order to ensure identifiability of the model, we restrict our attention to polynomially smooth, Sobolev-type densities f , with smoothness parameter β. In this context, we first provide a consistent estimation procedure for s. This estimator is then plugged-into three different procedures: estimation of the unknown density f , of the functional ∫