# 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.
Keywords :
Type de document :
Pré-publication, Document de travail
MAP5 2017-05. 2017
Domaine :

Littérature citée [32 références]

https://hal.archives-ouvertes.fr/hal-01449799
Contributeur : Fabienne Comte <>
Soumis le : mercredi 4 octobre 2017 - 14:02:55
Dernière modification le : jeudi 31 mai 2018 - 09:12:02

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LaguerreHermiteInverse10_17.pd...
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• HAL Id : hal-01449799, version 2

### Citation

Fabienne Comte, Valentine Genon-Catalot. Laguerre and Hermite bases for inverse problems . MAP5 2017-05. 2017. 〈hal-01449799v2〉

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