Nearest neighbours to nearest neighbours

Abstract

The distribution of the distance from an arbitrary point to the nth nearest event within a two-dimensional Poisson process is well known and appears frequently in the literature. These and other nearest neighbour distances have often been used for analyses of two-dimensional Poisson processes and other spatial patterns; such analyses may include testing for randomness and the robust estimation of density. In this article nearest neighbour distances are not considered but the related problem of nearest neighbours to nearest neighbours. The probability is found that the mtth nearest neighbour to an arbitrary point within a two-dimensional Poisson process is the kth nearest neighbour to the nth nearest neighbour of the original point. Similar nearest neighbour relationships are also considered. Two possible users are outlined for results obtained.