Open Access
July, 1977 Consistent Nonparametric Regression
Charles J. Stone
Ann. Statist. 5(4): 595-620 (July, 1977). DOI: 10.1214/aos/1176343886


Let $(X, Y)$ be a pair of random variables such that $X$ is $\mathbb{R}^d$-valued and $Y$ is $\mathbb{R}^{d'}$-valued. Given a random sample $(X_1, Y_1), \cdots, (X_n, Y_n)$ from the distribution of $(X, Y)$, the conditional distribution $P^Y(\bullet \mid X)$ of $Y$ given $X$ can be estimated nonparametrically by $\hat{P}_n^Y(A \mid X) = \sum^n_1 W_{ni}(X)I_A(Y_i)$, where the weight function $W_n$ is of the form $W_{ni}(X) = W_{ni}(X, X_1, \cdots, X_n), 1 \leqq i \leqq n$. The weight function $W_n$ is called a probability weight function if it is nonnegative and $\sum^n_1 W_{ni}(X) = 1$. Associated with $\hat{P}_n^Y(\bullet \mid X)$ in a natural way are nonparametric estimators of conditional expectations, variances, covariances, standard deviations, correlations and quantiles and nonparametric approximate Bayes rules in prediction and multiple classification problems. Consistency of a sequence $\{W_n\}$ of weight functions is defined and sufficient conditions for consistency are obtained. When applied to sequences of probability weight functions, these conditions are both necessary and sufficient. Consistent sequences of probability weight functions defined in terms of nearest neighbors are constructed. The results are applied to verify the consistency of the estimators of the various quantities discussed above and the consistency in Bayes risk of the approximate Bayes rules.


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Charles J. Stone. "Consistent Nonparametric Regression." Ann. Statist. 5 (4) 595 - 620, July, 1977.


Published: July, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0366.62051
MathSciNet: MR443204
Digital Object Identifier: 10.1214/aos/1176343886

Primary: 62G05
Secondary: 62H30

Keywords: approximate Bayes rules , conditional quantities , consistency in Bayes risk , multiple classification , nearest neighbor rules , nonparametric estimators , prediction , regression function

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 4 • July, 1977
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