ESTIMATION OF CONVOLUTION IN THE MODEL WITH NOISE

Abstract. We investigate the estimation of the ℓ-fold convolution of the density of an unob-served variable X from n i.i.d. observations of the convolution model Y = X + ε. We first assume that the density of the noise ε is known and define nonadaptive estimators, for which we provide bounds for the mean integrated squared error (MISE). In particular, under some smoothness assumptions on the densities of X and ε, we prove that the parametric rate of con-vergence 1/n can be attained. Then we construct an adaptive estimator using a penalization approach having similar performances to the nonadaptive 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