Deconvolution with Estimated Characteristic Function of the Errors

We study the following model of deconvolution Y = X+ε with 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 . The aim of the paper is to estimate f without knowing the density fε of the εj . We first define an estimator, for which we provide bounds for the pointwise and 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.