Estimation of a fold convolution in additive noise model with compactly supported noise density

Abstract

Consider the model Y = X + Z , where Y is an observable random variable, X is an unobservable random variable with unknown density f , and Z is a random noise independent of X . The density g of Z is known exactly and assumed to be compactly supported. We are interested in estimating the m- fold convolution f_m=f*...*f on the basis of independent and identically distributed (i.i.d.) observations Y₁,..,Y_n drawn from the distribution of Y . Based on the observations as well as the ridge-parameter regularization method, we propose an estimator for the function f_m depending on two regularization parameters in which a parameter is given and a parameter must be chosen. The proposed estimator is shown to be consistent with respect to the mean integrated squared error under some conditions of the parameters. After that we derive a convergence rate of the estimator under some additional regular assumptions for the density f .