Integrated Square Error Properties of Kernel Estimators of Regression Functions

Abstract

Weak laws of large numbers and central limit theorems are proved for integrated square error of kernel estimators of regression functions. The regressor is assumed to take values in $\mathbb{R}^p$, and the regressand, $X$, to be real valued. It is shown that in many cases, integrated square error is asymptotically normally distributed and independent of the $X$-sample. As an application, a test for the regression function (such as that proposed by Konakov) is seen to be asymptotically independent of an arbitrary test based on the $X$-sample. The proofs involve martingale methods.