On the Gini Mean Difference Test for Circular Data

In this paper, we propose a new test of uniformity on the circle based on the Gini mean difference of the sample arc-lengths. These sample arc-lengths, which are the gaps between successive observations on the circumference of the circle, are analogous to sample spacings on the real line. The Gini mean difference, which compares these arc-lengths between themselves, is analogous to Rao’s spacings statistic, which has been used to test the uniformity of circular data. We obtain both the exact and asymptotic distributions of the Gini mean difference arc-lengths test, under the null hypothesis of circular uniformity. We also provide a table of upper percentile values of the exact distribution for small to moderate sample sizes. Illustrative examples in circular data analysis are also given. It is shown that a generalized Gini mean difference test has better asymptotic efficiency than the corresponding generalized Rao’s test in the sense of Pitman asymptotic relative efficiency.