Tests for One-dimensional Nearest Neighbors Clusters

Abstract

A finite population of points is generated on the real line by a Poisson process. From each point an arrow is drawn to its nearest neighbor (NN). In this way clusters of points are generated which are connected by arrows. (1999) studied the distribution of the number (M) of NN clusters. Because too small or too large values of M may indicate deviations from the null hypothesis of no clustering, tests for the detection of clusters can be based on the distribution of M. Such tests are of interest in epidemiology if one wants to know whether the times of occurrence of a certain disease cluster. We observe that the problem formulated (1999) is equivalent to the problem of deriving the distribution of the number of peaks in randomization problems which was studied by (1962) and others. Because the distribution of M has to be calculated by means of a recurrence equation, which makes the performance of the tests mentioned above difficult, we propose some bounds for the P -values and compare these with the exact distribution and a normal approximation.