An Infinite-Dimensional Approximation for Nearest Neighbor Goodness of Fit Tests

Abstract

Let $X_1, \cdots, X_n$ be i.i.d. $\mathbb{R}^m$-valued observations from a bounded density $g(x)$ continuous on its support. Let $W_i = \exp\{- ng(X_i)V(R_i)\}, i = 1, \cdots, n$, where $V(R_i)$ is the volume of the nearest neighbor sphere around $X_i$, and let $w(x)$ be any bounded continuous weight function on $\mathbb{R}^m$. An infinite-dimensional approximation to the asymptotic form of the weighted empirical distribution function of the $W_i$'s is presented. The distributions of quadratic functionals of the limiting normalized weighted e.d.f. are found and tabulated for $m = \infty$ and $m = 1$ and compared with finite $m > 1$. Monte Carlo results are given for $n, m < \infty$.