On the Number of Reflexive and Shared Nearest Neighbor Pairs in One-Dimensional Uniform Data

TL;DR

This paper analyzes the distribution and asymptotic behavior of reflexive and shared nearest neighbor pairs in one-dimensional uniform data, providing exact formulas, recurrence relations, and limit theorems.

Contribution

It derives exact expectations, variances, and a recurrence relation for the number of NN pairs, introducing a novel approach for their probability distribution.

Findings

Exact formulas for expected values and variances of R_n and Q_n

A recurrence relation for the pmf of R_n

Asymptotic normality and strong law of large numbers for R_n and Q_n

Abstract

For a random sample of points in $\mathbb{R}$, we consider the number of pairs whose members are nearest neighbors (NN) to each other and the number of pairs sharing a common NN. The first type of pairs are called reflexive NNs whereas latter type of pairs 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 as $R_n$ 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 of $R_n$. Our approach is a novel method for finding the pmf of $R_n$ and agrees with the results in literature. We also present SLLN and CLT results for both $R_n$ and $Q_n$ as $n$ goes to infinity.