Kernel Smoothing in Partial Linear Models

TLDR

Kernel smoothing is investigated within partial linear models, where the response is expressed as a linear component plus an unknown smooth function of a covariate. The authors compare two estimation approaches: a partial smoothing spline method and a partial residual analysis method for jointly estimating the linear coefficient and the smooth function. They derive asymptotic bias and variance for both methods, showing that the partial residual approach yields lower bias without increasing variance, and illustrate the results through ANCOVA applications and examples.

Abstract

SUMMARY Kernel smoothing is studied in partial linear models, i.e. semiparametric models of the form yi=ξi′β+f(ti)+εi(1⩽i⩽n), where the ξi are fixed known p vectors, β is an unknown vector parameter and f is a smooth but unknown function. Two methods of estimating β and f are considered, one related to partial smoothing splines and the other motivated by partial residual analysis. Under suitable assumptions, the asymptotic bias and variance are obtained for both methods, and it is shown that estimating β by partial residuals results in improved bias with no asymptotic loss in variance. Application to analysis of covariance is made, and several examples are presented.