The Annals of Statistics

Integrated Square Error Properties of Kernel Estimators of Regression Functions

Peter Hall

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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.

Article information

Ann. Statist., Volume 12, Number 1 (1984), 241-260.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62G20: Asymptotic properties
Secondary: 62E20: Asymptotic distribution theory 60F05: Central limit and other weak theorems

Central limit theorem integrated square error kernel estimator law of large numbers multivariate nonparametric regression test for regression


Hall, Peter. Integrated Square Error Properties of Kernel Estimators of Regression Functions. Ann. Statist. 12 (1984), no. 1, 241--260. doi:10.1214/aos/1176346404.

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