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 Y-i = XiUi, Z(i) = X-i + Sigma(i), where the X-i's are i.i.d. with unknown density f, the Sigma(i)'s are i.i.d. with known density f(Sigma), the U-i's are i.i.d. with uniform density on [0, 1]. The sequences (X-i), (U-i), (Sigma(i)) are independent. We define projection estimators of f in the two cases of indirect observations of (X-1, ..., X-n), and we give upper bounds for their L-2-risks on specific Sobolev-Laguerre or Sobolev-Hermite spaces. Data-driven procedures are described and proved to perform automatically the bias-variance compromise. (C) 2018 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.