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In this paper we propose a convolution estimator for estimating the densityof a response variable that employs an underlying multiple regression frameworkto enhance the accuracy of density estimates through the incorporation ofauxiliary information. Suppose we have a sample consisting of $N$ complete caseobservations of a response variable and an associated set of covariates, alongwith an additional sample consisting of $M$ observations of the covariatesonly. We show that the mean square error of the multiple regression-enhancedconvolution estimator converges as $O(N^{-1})$ towards zero, and moreover, fora 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 ofa convolution estimator with respect to the amount of additional covariateinformation has been established. In contrast to convolution estimators basedon the Nadaraya-Watson estimator for a nonlinear regression model, the multipleregression-enhanced convolution estimator proposed in this paper does notsuffer from the curse of dimensionality. It is particularly useful forscenarios in which one wants to estimate the density of a response variablethat is challenging to measure, while being in possession of a large amount ofadditional covariate information. In fact, an application of this type from thefield of ophthalmology motivated our work in this paper.

A Multiple Regression-Enhanced Convolution Estimator for the Density of a Response Variable in the Presence of Additional Covariate Information