Test for Uniformity of Exchangeable Random Variables on the Circle

Abstract

We are motivated by our laboratory experiment on the flocking behavior of termites. To test for the existence of flocking behavior, we revisit the problem to test uniform samples (with the samples uniformly distributed) on the circle. Unlike most existing works, we assume that the samples are exchangeably dependent. We consider the class of normalized infinitely divisible distributions for the spacings of the samples, which form uniform samples on the circle. To test the uniformity, we study a test (Kuiper’s test) based on spacings of the samples and compute the asymptotic null distribution of the test statistic as the scaled Kolmogorov distribution. We apply the procedure to our experimental data and justify the flocking behavior of termites.