A Multi-Sample Uniform Scores Test on a Circle and its Parametric Competitor

Summary We propose a c-sample non-parametric test W on a circle based on uniform scores. The test can be regarded as an extension of Wheeler and Watson's test (1964). This test is constructed with the help of a variant of a c-sample test for von Mises populations under the shift-type of alternatives. We indicate the relationship of the W-test with a test given by Mardia (1970). This result solves the problem of critical values of this test and of approximations to the null distribution of W. The asymptotic distribution of this test and of its parametric competitor test S are obtained. Bahadur's efficiency of the W-test relative to the S-test is derived and some important particular cases are considered. The tests are applied to a numerical example.