Adaptive deconvolution of linear functionals on the nonnegative real line

Abstract

In this paper we consider the convolution model Z=X+Y with X of unknown density f, independent of Y, when both random variables are nonnegative. Our goal is to estimate linear functionals of f such as 〈ψ,f〉 for a known function ψ assuming that the distribution of Y is known and only Z is observed. We propose an estimator of 〈ψ,f〉 based on a projection estimator of f on Laguerre spaces, present upper bounds on the quadratic risk and derive the rate of convergence in function of the smoothness of f,g and ψ. Then we propose a nonparametric data driven strategy, inspired Goldenshluger and Lepski (2011) method to select a relevant projection space. This methodology permits to estimate the cumulative distribution function of X for instance. In addition it is adapted to the pointwise estimation of f. We illustrate the good performance of the new method through simulations. We also test a new approach for choosing the tuning parameter in Goldenshluger–Lepski data driven estimators following ideas developed in Lacour and Massart (2015).