On Efficient Estimation in Regression Models

Abstract

In this paper we consider the regression model with smooth regression function and smooth error and covariate distributions. We study how well one can estimate functionals of the regression function which may also depend on the distribution of the covariate. This is done by deriving the efficient influence functions of least dispersed regular estimators of such functionals under various assumptions on the parameters of our model. Then we demonstrate how efficient estimates can be constructed. We provide a general procedure for constructing efficient estimates that relies on appropriate auxiliary estimates. We illustrate the usefulness of this procedure by constructing efficient estimates for various parametric, nonparametric and semiparametric models.