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March, 1983 An Infinite-Dimensional Approximation for Nearest Neighbor Goodness of Fit Tests
Mark F. Schilling
Ann. Statist. 11(1): 13-24 (March, 1983). DOI: 10.1214/aos/1176346052


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 $n, m < \infty$.


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Mark F. Schilling. "An Infinite-Dimensional Approximation for Nearest Neighbor Goodness of Fit Tests." Ann. Statist. 11 (1) 13 - 24, March, 1983.


Published: March, 1983
First available in Project Euclid: 12 April 2007

zbMATH: 0532.62076
MathSciNet: MR684859
Digital Object Identifier: 10.1214/aos/1176346052

Primary: 62M99
Secondary: 62E20 , 62G10 , 62H15

Keywords: Empirical distribution function , infinite-dimensional approximation , nearest neighbor , quadratic functional

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 1 • March, 1983
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