## The Annals of Statistics

### Integrated Square Error Properties of Kernel Estimators of Regression Functions

Peter Hall

#### 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

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