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November, 1981 On the Almost Everywhere Convergence of Nonparametric Regression Function Estimates
Luc Devroye
Ann. Statist. 9(6): 1310-1319 (November, 1981). DOI: 10.1214/aos/1176345647


Let $(X, Y), (X_1, Y_1), \cdots, (X_n, Y_n)$ be independent identically distributed random vectors from $R^d \times R$, and let $E(|Y|^p) < \infty$ for some $p \geq 1$. We wish to estimate the regression function $m(x) = E(Y \mid X = x)$ by $m_n(x)$, a function of $x$ and $(X_1, Y_1), \cdots, (X_n, Y_n)$. For large classes of kernel estimates and nearest neighbor estimates, sufficient conditions are given for $E\{|m_n(x) - m(x)|^p\} \rightarrow 0$ as $n \rightarrow \infty$, almost all $x$. No additional conditions are imposed on the distribution of $(X, Y)$. As a by-product, just assuming the boundedness of $Y$, the almost sure convergence to 0 of $E\{|m_n(X) - m(X)\| X_1, Y_1, \cdots, X_n, Y_n\}$ is established for the same estimates. Finally, the weak and strong Bayes risk consistency of the corresponding nonparametric discrimination rules is proved for all possible distributions of the data.


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Luc Devroye. "On the Almost Everywhere Convergence of Nonparametric Regression Function Estimates." Ann. Statist. 9 (6) 1310 - 1319, November, 1981.


Published: November, 1981
First available in Project Euclid: 12 April 2007

zbMATH: 0477.62025
MathSciNet: MR630113
Digital Object Identifier: 10.1214/aos/1176345647

Primary: 62G05

Keywords: kernel estimate , nearest neighbor rule , nonparametric discrimination , regression function , universal consistency

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 6 • November, 1981
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