Root n consistent and optimal density estimators for moving average processes

The marginal density of a first order moving average process can be written as a convolution of two innovation densities. Saavedra & Cao [ Can. J. Statist. ( 2000), 28, 799] propose to estimate the marginal density by plugging in kernel density estimators for the innovation densities, based on estimated innovations. They obtain that for an appropriate choice of bandwidth the variance of their estimator decreases at the rate 1/n. Their estimator can be interpreted as a specific U-statistic. We suggest a slightly simplified U-statistic as estimator of the marginal density, prove that it is asymptotically normal at the same rate, and describe the asymptotic variance explicitly. We show that the estimator is asymptotically efficient if no structural assumptions are made on the innovation density. For innovation densities known to have mean zero or to be symmetric, we describe improvements of our estimator which are again asymptotically efficient.