Adaptive deconvolution of linear functionals on the nonnegative real line

• We propose a non-parametric estimation of linear functionals of the density in the convolution model on the nonnegative real line.
• The methodology permits to estimate cumulative and probability distribution functions as well as Laplace transform.
• We also build a data-driven strategy to estimate linear functionals in this model.

In this paper we consider the convolution model Z=X+YZ=X+Y with XX of unknown density ff, independent of YY, when both random variables are nonnegative. Our goal is to estimate linear functionals of ff such as 〈ψ,f〉〈ψ,f〉 for a known function ψψ assuming that the distribution of YY is known and only ZZ is observed. We propose an estimator of 〈ψ,f〉〈ψ,f〉 based on a projection estimator of ff 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 ψψ. 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 XX for instance. In addition it is adapted to the pointwise estimation of ff. 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).