Use of regression functions for improved estimation of means

Abstract

Suppose one has a random sample from a distribution F Y,X, the goal being to estimate μ, the unconditional mean of Y . If the regression function of Y on X is of a known parametric form, one might consider estimating μ by the average of the estimated regression function values at the sample points. It is shown here that such an estimator can offer substantial improvement over the sample mean of Y , and that in no case does the former estimator have larger asymptotic variance than the latter estimator. Applications of this asymptotic distribution theory to the double-sampling setting are also given.