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

We consider the model: Y = X+ , where X and are independent random variables. The density of is known whereas the one of X is a finite mixture with unknown components. Considering the ”ordinary smooth case” on the density of , 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 taking the minimax approach under the mean integrated squared error over Besov balls. We prove that the adaptive one attains a sharp rate of convergence.