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

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.