On the fraction of random points by specified nearest-neighbour interrelations and degree of attraction
- 1 December 1987
- journal article
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 19 (4) , 873-895
- https://doi.org/10.2307/1427106
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
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