Ordering Distributions on the Circle with Respect to Uniformity

摘要

Orderings (denoted <<) for probability distributions on the circle are introduced for which mu1 << mu2 means roughly that mu1 is more uniform than mu2, or that mu2 is more clumped than mu1. Somewhat more precisely, mu1 << mu2 if the random variable mu1(A sub s) is 'less variable' than the random variable mu2(A sub s) for all s where A sub s denotes a random arc of length s distributed uniformly on the circle. Some properties of the orderings are explored and applications are presented to random coverage problems on the circle, the distribution of spacings and the analysis of directional data. Keywords: Spacings, Random arcs, Tests of uniformity, Directional data, Order statistics, Majorization, Convex ordering. (jhd)