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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 38, Fasc. 1,
pages 123 - 137
 

ON THE NUMBER OF REFLEXIVE AND SHARED NEAREST NEIGHBOR PAIRS IN ONE-DIMENSIONAL UNIFORM DATA

Selim Bahadir
Elvan Ceyhan

Abstract: For a random sample of points in ℝ  , 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 n 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 R
 n  and Q
  n  , respectively. We derive the exact forms of the expected value and the variance for both R
  n  and Q
  n  , and derive a recurrence relation for R
  n  which may also be used to compute the exact probability mass function (pmf) of R
 n  . Our approach is a novel method for finding the pmf of R
 n  and agrees with the results in the literature. We also present SLLN and CLT results for both R
  n  and Q
 n  as n goes to infinity.

2010 AMS Mathematics Subject Classification: Primary: 05A05; Secondary: 05C20, 05C80, 60F05.

Keywords and phrases: Asymptotic normality, central limit theorem, exact distribution, law of large numbers, nearest neighbor graphs and digraphs, random permutation.

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