Nonparametric Laguerre estimation in the multiplicative censoring model

Abstract

We study the model $Y_{i}=X_{i}U_{i},\;i=1,\ldots,n$ where the $U_{i}$’s are i.i.d. with $\beta(1,k)$ density, $k\ge1$, $k$ integer, the $X_{i}$’s are 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.