The uniform distribution for directional and cyclic data. The probability density function f is constant for directions (in radians). For a data set θ1, θ2,…, θn, a test of circular uniformity amounts to a test of the null hypothesis, H0, that there is no preferred direction, with the alternative being that the data come from some unimodal circular distribution. The Rayleigh test, introduced by Lord Rayleigh in 1880, uses the test statistic R2 given by . Under H0 the approximate probability of observing a value≥R2 is given by . For very large values of n, T=2R2/n is an observation from an approximate chi-squared distribution with two degrees of freedom.
As an alternative, the Ajne test (suggested by Bjorn Ajne in 1968) uses as its test statistic the number of observations contained within the semicircle for which this number is least.
Subjects: Probability and Statistics.