Nonparametric Laguerre estimation in the multiplicative censoring model

Denis Belomestny (1) , Fabienne Comte (2) , Valentine Genon-Catalot (2)

1 Universität Duisburg-Essen = University of Duisburg-Essen [Essen]
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

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.

Mots clés: Adaptive estimation; Lower bounds; Model selection; Multiplicative censoring; k-monotone densities; Projection estimator

Domaines: Statistiques [math.ST]

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