Efficiency of linear regression estimators based on presmoothing

Abstract

Consider the estimation of the regression parameters in the usual linear model. For design densities with infinite support, it has been shown by Faraldo Foca and González Manteiga(1) that it is possible to modify the classical least squares procedures and to obtain estimators for the regression parameters whose MSE's(mean squared errors) are smaller than those of the usual least squared estimators. The modification consists of presmoothing the response variables by a kernel estimator of the regression function. These authors also show that the gain in efficiency is not possible for a design density with compact support. We show that in this case local linear fitting automatically corrects the bias in the endpoints of the (design density) support. We demonstrate on a theoretical basis how this inefficiency problem can be rectified in the compact design case: we prove that presmoothing with boundary kernels studied in Müller(2) and Müler and Wang(3) leads to regression estimators which are superior over the least squares estimators. A very careful analytic treatment is needed to arrive at these asymptotic results.