ON THE NUMBER OF REFLEXIVE AND SHARED NEAREST NEIGHBOR
PAIRS IN ONE-DIMENSIONAL UNIFORM DATA
Abstract: For a random sample of points in , we consider the number of pairs whose members
are nearest neighbors (NNs) to each other and the number of pairs sharing a common NN.
The pairs of the first type are called reflexive NNs, whereas the pairs of the latter type are
called shared NNs. In this article, we consider the case where the random sample of
size is from the uniform distribution on an interval. We denote the number of
reflexive NN pairs and the number of shared NN pairs in the sample by and ,
respectively. We derive the exact forms of the expected value and the variance for both
and , and derive a recurrence relation for which may also be used to
compute the exact probability mass function (pmf) of . Our approach is a novel
method for finding the pmf of and agrees with the results in the literature. We
also present SLLN and CLT results for both and as goes to infinity.
2010 AMS Mathematics Subject Classification: Primary: 05A05; Secondary: 05C20,
Keywords and phrases: Asymptotic normality, central limit theorem, exact
distribution, law of large numbers, nearest neighbor graphs and digraphs, random