Laguerre estimation for k-monotone densities observed with noise

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 models Zi = Yi + Vi, Yi = XiUi, i = 1,. .. , n where the Vi's are nonnegative, i.i.d. with known density fV , the Ui's are i.i.d. with β(1, k) density, k ≥ 1, the Xi's are i.i.d., nonnegative with unknown density f. The sequences (Xi), (Ui), (Vi) are independent. We aim at estimating f on R + in the three cases of direct observations (X1,. .. , Xn), observations (Y1,. .. , Yn), observations (Z1,. .. , Zn). We propose projection estimators using a Laguerre basis and give upper bounds on the L 2-risks on specific Sobolev-Laguerre spaces. Lower bounds matching with the upper bounds are proved in the case of direct observation of X and in the case of observation of Y. A general data-driven procedure is described and proved to perform automatically the bias variance compromise. 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. Laguerre estimation for k-monotone densities observed with noise. 2016. ⟨hal-01252143v1⟩