  In this paper we propose a convolution estimator for estimating the density
of a response variable that employs an underlying multiple regression framework
to enhance the accuracy of density estimates through the incorporation of
auxiliary information. Suppose we have a sample consisting of $N$ complete case
observations of a response variable and an associated set of covariates, along
with an additional sample consisting of $M$ observations of the covariates
only. We show that the mean square error of the multiple regression-enhanced
convolution estimator converges as $O(N^{-1})$ towards zero, and moreover, for
a large fixed $N$, that the mean square error converges as $O(M^{-4/5})$
towards an $O(N^{-1})$ constant. This is the first time that the convergence of
a convolution estimator with respect to the amount of additional covariate
information has been established. In contrast to convolution estimators based
on the Nadaraya-Watson estimator for a nonlinear regression model, the multiple
regression-enhanced convolution estimator proposed in this paper does not
suffer from the curse of dimensionality. It is particularly useful for
scenarios in which one wants to estimate the density of a response variable
that is challenging to measure, while being in possession of a large amount of
additional covariate information. In fact, an application of this type from the
field of ophthalmology motivated our work in this paper.
