# On the number of reflexive and shared nearest neighbor pairs in one-dimensional uniform data

## DOI:

https://doi.org/10.19195/0208-4147.38.1.7## Słowa kluczowe:

Asymptotic normality, central limit theorem, exact distribution, law of large numbers, nearest neighbor graphs and digraphs, random permutation## Abstrakt

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* Rn* and *Qn*, respectively. We derive the exact forms of the expected value and the variance for both *Rn* and *Qn*, and derive a recurrence relation for *Rn* which may also be used to compute the exact probability mass function pmf of* Rn*. Our approach is a novel method for finding the pmf of Rn and agrees with the results in the literature. We also present SLLN and CLT results for both *Rn* and *Qn* as n goes to infinity.