Wavelet density estimators for the deconvolution of a component from a mixture

Abstract

We consider the model: Y = X + ε, where X and ε are independent random variables. The density of ε is known whereas the one of X is a finite mixture with unknown components. Considering the “ordinary smooth case” on the density of ε, we want to estimate a component of this mixture. To reach this goal, we develop two wavelet estimators: a nonadaptive based on a projection and an adaptive based on a hard thresholding rule. We evaluate their performances by considering the mean integrated squared error over Besov balls. We prove that the adaptive one attains a sharp rate of convergence.