Density estimators for the convolution of discrete and continuous random variables

Suppose we have independent observations of a pair of independent random variables, one with a density and the other discrete. The sum of these random variables has a density, which can be estimated by an ordinary kernel estimator. Since the two components are independent, we can write the density as a convolution and alternatively estimate it by a convolution of a kernel estimator of the continuous component with an empirical estimator of the discrete component. We show that for a given kernel and optimal bandwidth, this estimator has the same rate as the first estimator, and the same asymptotic bias, but a much smaller asymptotic variance. We also show how pointwise constraints on derivatives of the density of the continuous component can be used to improve our estimator of the convolution density.