Data-driven density estimation in the presence of additive noise with unknown distribution

We study the model Y = X + epsilon. We assume that we have at our disposal independent identically distributed observations Y(1), ... ,Y(n) and epsilon(-1), ... ,epsilon(-M). The (X(j))(1 <= j <= n) are independent identically distributed with density f, independent of the (epsilon(j))(1 <= j <= n), independent identically distributed with density f(epsilon). The aim of the paper is to estimate f without knowing f(epsilon). We first define an estimator, for which we provide bounds for the integrated L(2)-risk. We consider ordinary smooth and supersmooth noise epsilon with regard to ordinary smooth and supersmooth densities f. Then we present an adaptive estimator of the density of f. This estimator is obtained by penalization of a projection contrast and yields to model selection. Lastly, we present simulation experiments to illustrate the good performances of our estimator and study from the empirical point of view the importance of theoretical constraints.