Laguerre and Hermite bases for inverse problems

Abstract : We present inverse problems of nonparametric statistics which have a performing and smart solution using projection estimators on bases of functions with non compact support, namely, a Laguerre basis or a Hermite basis. The models are $Y_i=X_iU_i,\;Z_i=X_i+\Sigma_i,$ where the $X_i$'s are {\em i.i.d.} with unknown density $f$, the $\Sigma_i$'s are {\em i.i.d.} with known density $f_\Sigma$, the $U_i$'s are {\em 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 $(X_1, \dots, X_n)$, and we give upper bounds for their ${\mathbb L}^2$-risks on specific Sobolev-Laguerre or Sobolev-Hermite spaces. Data-driven procedures are described and proved to perform automatically the bias variance compromise.
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Fabienne Comte, Valentine Genon-Catalot. Laguerre and Hermite bases for inverse problems . Journal of the Korean Statistical Society, Elsevier, 2018, 47 (3), pp.273-296. ⟨10.1016/j.jkss.2018.03.001⟩. ⟨hal-01449799v2⟩

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