## On the Gini Mean Difference Test for Circular Data

**Authors:** David D. Tung, S. Rao Jammalamadaka

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.

**Comments:** 14 Pages.

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### Submission history

[v1] 23 Jul 2010

[v2] 16 Aug 2010

[v3] 2012-01-11 19:14:18

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