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May, 1977 The Strong Uniform Consistency of Nearest Neighbor Density Estimates
Luc P. Devroye, T. J. Wagner
Ann. Statist. 5(3): 536-540 (May, 1977). DOI: 10.1214/aos/1176343851

Abstract

Let $X_1,\cdots, X_n$ be independent, identically distributed random vectors with values in $\mathbb{R}^d$ and with a common probability density $f$. If $V_k(x)$ is the volume of the smallest sphere centered at $x$ and containing at least $k$ of the $X_1,\cdots, X_n$ then $f_n(x) = k/(nV_k(x))$ is a nearest neighbor density estimate of $f$. We show that if $k = k(n)$ satisfies $k(n)/n \rightarrow 0$ and $k(n)/\log n \rightarrow \infty$ then $\sup_x|f_n(x) - f(x)|\rightarrow 0$ w.p. 1 when $f$ is uniformly continuous on $\mathbb{R}^d$.

Citation

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Luc P. Devroye. T. J. Wagner. "The Strong Uniform Consistency of Nearest Neighbor Density Estimates." Ann. Statist. 5 (3) 536 - 540, May, 1977. https://doi.org/10.1214/aos/1176343851

Information

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

zbMATH: 0367.62061
MathSciNet: MR436442
Digital Object Identifier: 10.1214/aos/1176343851

Subjects:
Primary: 60F15
Secondary: 62G05

Keywords: consistency , multivariate density estimation , Nonparametric density estimation , uniform consistency

Rights: Copyright © 1977 Institute of Mathematical Statistics

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Vol.5 • No. 3 • May, 1977
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