Consistency Properties of Nearest Neighbor Density Function Estimators

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.