Data Driven Density Estimation in Presence of Additive Noise with Unknown Distribution

We study the following model: Y = X + ε. We assume that we have at our disposal i.i.d. observations Y1, . . . , Yn and ε−1, . . . , ε−M . The (Xj)1≤j≤n are i.i.d. with density f , independent of the (εj)1≤j≤n, i.i.d. with density fε. The aim of the paper is to estimate f without knowing fε. We first define an estimator, for which we provide bounds for the integrated L-risk. We consider ordinary smooth and supersmooth noise ε 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.