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December, 1984 Distribution-Free Pointwise Consistency of Kernel Regression Estimate
Wlodzimierz Greblicki, Adam Krzyzak, Miroslaw Pawlak
Ann. Statist. 12(4): 1570-1575 (December, 1984). DOI: 10.1214/aos/1176346815

Abstract

An estimate $\sum^n_{i=1} Y_iK((x - X_i)/h)/\sum^n_{j=1} K((x - X_j)/h)$, calculated from a sequence $(X_1, Y_1), \cdots, (X_n, Y_n)$ of independent pairs of random variables distributed as a pair $(X, Y)$, converges to the regression $E\{Y\mid X = x\}$ as $n$ tends to infinity in probability for almost all $(\mu) x \in R^d$, provided that $E|Y| < \infty, h \rightarrow 0$ and $nh^d \rightarrow \infty$ as $n \rightarrow \infty$. The result is true for all distributions $\mu$ of $X$. If, moreover, $|Y| \leq \gamma < \infty$ and $nh^d/\log n \rightarrow \infty$ as $n \rightarrow \infty$, a complete convergence holds. The class of applicable kernels includes those having unbounded support.

Citation

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Wlodzimierz Greblicki. Adam Krzyzak. Miroslaw Pawlak. "Distribution-Free Pointwise Consistency of Kernel Regression Estimate." Ann. Statist. 12 (4) 1570 - 1575, December, 1984. https://doi.org/10.1214/aos/1176346815

Information

Published: December, 1984
First available in Project Euclid: 12 April 2007

zbMATH: 0551.62025
MathSciNet: MR760711
Digital Object Identifier: 10.1214/aos/1176346815

Subjects:
Primary: 62G05

Rights: Copyright © 1984 Institute of Mathematical Statistics

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Vol.12 • No. 4 • December, 1984
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