Integrated Square Error Properties of Kernel Estimators of Regression Functions
The Annals of Statistics · 1984 · Volume 12 · Issue 1
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.
Keywords
60F05, 62E20, 62G20, central limit theorem, integrated square error, Kernel estimator, Law of Large Numbers, multivariate, nonparametric, regression, test for regression
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