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On the fraction of random points by specified nearest-neighbour interrelations and degree of attraction

Published online by Cambridge University Press:  01 July 2016

Norbert Henze
Norbert Henze*
Affiliation:
University of Hannover
*
∗Postal address: Institut für Mathematische Stochastik, Universität Hannover, Welfengarten 1, D-3000 Hannover 1, W. Germany.
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Abstract

Let Z1, …, Zn be i.i.d. random vectors (‘points') defined in having common density f(x) that is assumed to be continuous almost everywhere. For a fixed but otherwise arbitrary norm |.| on , consider the fraction Vn of those points Z1, …, Zn that are the lth nearest neighbour (with respect to |.|) to their own kth nearest neighbour, and write Sn for the fraction of points that are the nearest neighbour of exactly k other points. We derive the stochastic limits of Vn and Sn, as n tends to∞, and show how the results may be applied to the multivariate non-parametric two-sample problem.

Keywords

NEAREST NEIGHBOURSGEOMETRIC PROBABILITYPOISSON PROCESSMULTIVARIATE TWO-SAMPLE PROBLEM
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 4 , December 1987 , pp. 873 - 895
Copyright
Copyright © Applied Probability Trust 1987 

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