LAGUERRE ESTIMATION FOR k-MONOTONE DENSITIES OBSERVED WITH NOISE D. BELOMESTNY (1) , F. COMTE (2) & V. GENON-CATALOT (3) Abstract. We study the models Zi = Yi + Vi ,Yi = Xi Ui ,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 inde- pendent. 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. (1) Duisburg-Essen University, email: denis.belomestny@uni-due.de, (2) Universit´ e Paris Descartes, MAP5, UMR CNRS 8145, email: fabienne.comte@parisdescartes.fr (3) Universit´ e Paris Descartes, MAP5, UMR CNRS 8145, valentine.genon-catalot@parisdescartes.fr. Keywords. Adaptive estimation. Lower bounds. Model selection. Multiplicative censoring. k-monotone densities. Projection estimator. MSC2010. 62G07 1. Introduction Consider observations Z 1 ,...,Z n such that (1) Z i = Y i + V i ,Y i = X i U i ,i =1,...,n. where X i ,U i ,V i are nonnegative random variables, (X i ) are i.i.d. with unknown density f , (U i ) are i.i.d.,(V i ) are i.i.d., U i ,V i have known densities and the sequences (X i ), (U i ),V i ) are independent. If V i = 0 and U i has uniform density on [0, 1], the model Z i = X i U i = Y i is called multiplicative censoring model and has been widely investigated in the past decades. As detailed in Vardi (1989), this model covers several important statistical problems, in particular estimation under monotonicity constraints. In this context, numerous papers deal with the estimation of f whether by nonparametric maximum likelihood (Vardi (1989), Vardi and Zhang (1992), Asgharian et al. (2012), by projection methods (Andersen and Hansen (2001), Abbaszadeh et al. (2012,2013)) or kernel methods (Brunel et al., (2015)). We mention that taking logarithm of the Y i allows a deconvolution method used by some authors (see e.g. van Es et al. (2003)): the method leads to estimate a distortion of f and cannot be straightforwardly extended to general model (1). Another approach to the above multiplicative deconvolution problem can be based on the Mellin transform technique (see Belomestny and Schoenmakers (2015) for a related problem). On the other hand, density estimation from noisy observations, i.e., estimation of the density f Y Date : January 13, 2016. 1