Adaptive Deconvolution on the Nonnegative Real Line

In this paper we consider the problem of adaptive density or survival function estimation in an additive model de ned by Z = X + Y with X independent of Y , when both random variables are nonnegative. We want to recover the distribution of X (density or survival function) through n observations of Z, assuming that the distribution of Y is known. This issue can be seen as the classical statistical problem of deconvolution which has been tackled in many cases using Fourier-type approaches. Nonetheless, in the present case the random variables have the particularity to be R+ supported. Knowing that, we propose a new angle of attack by building a projection estimator with an appropriate Laguerre basis. We present upper bounds on the mean squared integrated risk of our density and survival function estimators. We then describe a nonparametric adaptive strategy for selecting a relevant projection space. The procedures are illustrated with simulated data and compared to the performances of more classical deconvolution setting using a Fourier approach.