Estimation of convolution in the model with noise

We investigate the estimation of the l-fold convolution of the density of an unobserved variable X from n i.i.d. observations of the convolution model Y = X + epsilon. We first assume that the density of the noise e is known and define non-adaptive estimators, for which we provide bounds for the mean integrated squared error. In particular, under some smoothness assumptions on the densities of X and e, we prove that the parametric rate of convergence 1/n can be attained. Then, we construct an adaptive estimator using a penalisation approach having similar performances to the non-adaptive one. The price for its adaptivity is a logarithmic term. The results are extended to the case of unknown noise density, under the condition that an independent noise sample is available. Lastly, we report a simulation study to support our theoretical findings.