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

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.