The Annals of Statistics

Integrated Square Error Properties of Kernel Estimators of Regression Functions

Peter Hall

Full-text: Open access

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.

Article information

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176346404

Digital Object Identifier
doi:10.1214/aos/1176346404

Mathematical Reviews number (MathSciNet)
MR733511

Zentralblatt MATH identifier
0544.62036

JSTOR
links.jstor.org

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

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

Citation

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. https://projecteuclid.org/euclid.aos/1176346404


Export citation