Consistency Properties of Nearest Neighbor Density Function Estimators

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