Laguerre and Hermite bases for inverse problems

We present inverse problems of nonparametric statistics which have a smart solution using projection estimators on bases of functions with non compact support, namely, a Laguerre basis or a Hermite basis. The models are Yi=XiUi,Zi=Xi+Σi, where the Xi's are i.i.d. with unknown density f, the Σi's are i.i.d. with known density fΣ, the Ui's are i.i.d. with uniform density on [0,1]. The sequences (Xi),(Ui),(Σi) are independent. We define projection estimators of f in the two cases of indirect observations of (X1,…,Xn), and we give upper bounds for their L2-risks on specific Sobolev-Laguerre or Sobolev-Hermite spaces. Data-driven procedures are described and proved to perform automatically the bias-variance compromise.