A test of goodness-of-fit based on Gini's index of spacings

On the Gini Mean Difference Test for Circular Data

“…We provided a table of upper percentile values for this test, which will be useful to applied scientists employing it for circular data analysis. This new test extends the use of one by Jammalamadaka and Goria (2004) from the linear case to the circular case. On the basis of Pitman asymptotic relative efficiency, the generalized Gini mean difference test is asymptotically more efficient than the generalized Rao's test.…”

“…This test extends the Gini mean difference spacings test on the real line in Jammalamadaka and Goria (2004) to the circular case. 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 and provide a maximal-invariant under rotations so that all invariant tests have to be based on them.…”

“…The Gini mean difference spacings test was first proposed in Jammalamadaka and Goria (2004) for testing goodness-of-fit on the real line.…”

A multivariate uniformity test for the case of unknown support

“…They are calculated in a simple way in terms of differences between consecutive order statistics. The use of univariate spacings in the problem of testing uniformity is known since long time ago; see, e.g., Jammalamadaka and Goria (2004) and references therein.…”

“…Step 4: Now the proof is reduced to check (15) and (16). In order to prove (16) we will first show that…”