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January, 1977 Consistency Properties of Nearest Neighbor Density Function Estimators
David S. Moore, James W. Yackel
Ann. Statist. 5(1): 143-154 (January, 1977). DOI: 10.1214/aos/1176343747

Abstract

Let $X_1, X_2,\cdots$ be $R^p$-valued random variables having unknown density function $f$. If $K$ is a density on the unit sphere in $R^p, \{k(n)\}$ a sequence of positive integers such that $k(n) \rightarrow \infty$ and $k(n) = o(n)$, and $R(k, z)$ is the distance from a point $z$ to the $k(n)$th nearest of $X_1,\cdots, X_n$, then $f_n(z) = (nR(k, z)^p)^{-1} \sum K((z - X_i)/R(k, z))$ is a nearest neighbor estimator of $f(z).$ When $K$ is the uniform kernel, $f_n$ is an estimator proposed by Loftsgaarden and Quesenberry. The estimator $f_n$ is analogous to the well-known class of Parzen-Rosenblatt bandwidth estimators of $f(z)$. It is shown that, roughly stated, any consistency theorem true for the bandwidth estimator using kernel $K$ and also true for the uniform kernel bandwidth estimator remains true for $f_n$. In this manner results on weak and strong consistency, pointwise and uniform, are obtained for nearest neighbor density function estimators.

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David S. Moore. James W. Yackel. "Consistency Properties of Nearest Neighbor Density Function Estimators." Ann. Statist. 5 (1) 143 - 154, January, 1977. https://doi.org/10.1214/aos/1176343747

Information

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

zbMATH: 0358.60053
MathSciNet: MR426275
Digital Object Identifier: 10.1214/aos/1176343747

Subjects:
Primary: 60F15
Secondary: 62G05

Keywords: bandwidth estimators , multivariate density estimation , nearest neighbor estimators , Nonparametric density estimation , strong consistency , uniform consistency , weak consistency

Rights: Copyright © 1977 Institute of Mathematical Statistics

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Vol.5 • No. 1 • January, 1977
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