# Nonparametric Laguerre estimation in the multiplicative censoring model

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Abstract : We study the model $Y_i=X_iU_i, \; i=1, \ldots, n$ where the $U_i$'s are {\em i.i.d.} with $\beta(1,k)$ density, $k\ge 1$, the $X_i$'s are {\em i.i.d.}, nonnegative with unknown density $f$. The sequences $(X_i), (U_i),$ are independent. We aim at estimating $f$ on ${\mathbb R}^+$ from the observations $(Y_1, \dots, Y_n)$. We propose projection estimators using a Laguerre basis. A data-driven procedure is described in order to select the dimension of the projection space, which performs automatically the bias variance compromise. Then, we give upper bounds on the ${\mathbb L}^2$-risk on specific Sobolev-Laguerre spaces. Lower bounds matching with the upper bounds within a logarithmic factor are proved. The method is illustrated on simulated data.
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Cited literature [29 references]

https://hal.archives-ouvertes.fr/hal-01252143
Contributor : Fabienne Comte <>
Submitted on : Tuesday, May 24, 2016 - 10:27:34 PM
Last modification on : Saturday, April 11, 2020 - 2:01:38 AM

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### Citation

Denis Belomestny, Fabienne Comte, Valentine Genon-Catalot. Nonparametric Laguerre estimation in the multiplicative censoring model. Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2016, 10 (2), pp.3114-3152. ⟨10.1214/16-EJS1203⟩. ⟨hal-01252143v3⟩

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