The Strong Uniform Consistency of Nearest Neighbor Density Estimates

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 $\mathbb{R}^d$.