## The Annals of Statistics

- Ann. Statist.
- Volume 5, Number 4 (1977), 595-620.

### Consistent Nonparametric Regression

#### Abstract

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.

#### Article information

**Source**

Ann. Statist., Volume 5, Number 4 (1977), 595-620.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343886

**Digital Object Identifier**

doi:10.1214/aos/1176343886

**Mathematical Reviews number (MathSciNet)**

MR443204

**Zentralblatt MATH identifier**

0366.62051

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G05: Estimation

Secondary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]

**Keywords**

Regression function conditional quantities prediction multiple classification consistency in Bayes risk approximate Bayes rules nonparametric estimators nearest neighbor rules

#### Citation

Stone, Charles J. Consistent Nonparametric Regression. Ann. Statist. 5 (1977), no. 4, 595--620. doi:10.1214/aos/1176343886. https://projecteuclid.org/euclid.aos/1176343886