  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 the estimates through the incorporation of auxiliary
information. Suppose we have a sample of $N$ observations of a response
variable and an associated set of covariates, along with an additional
auxiliary sample featuring $M$ observations of the covariates only. We prove
that the mean square error of the multiple regression-enhanced estimator
converges as $O(N^{-1})$, and additionally, for a large fixed $N$, the mean
square error converges as $O(M^{-4/5})$ before eventually tailing off as a
saturation point is reached. Thus, while the incorporation of auxiliary
covariate information isn't quite as effective as incorporating more complete
case information, it nevertheless allows for significant improvements in
accuracy. In contrast to convolution estimators based on the Nadaraya-Watson
estimator for a nonlinear regression model, the convolution estimator proposed
herein utilizes the ordinary least squares estimator for a multiple linear
regression model. While this type of underlying estimator is not suited to
strongly nonlinear data, its strength lies in the fact that it allows the
multiple regression-enhanced convolution estimator to provide better
performance on data that is generally linear or well fit by low order
polynomials, since the ordinary least squares estimator estimator does not
suffer from the curse of dimensionality and does not require one to choose
hyperparameters. The estimator proposed in this paper is particularly useful
estimating the density of a response variable that is challenging to measure,
while being in possession of a large amount of auxiliary information. In fact,
an application of this type from the field of ophthalmology motivated our work
in this paper.
