Nonparametric symmetry tests for circular distributions

SUMMARY In this paper the locally most powerful rank test for testing the symmetry axis of a symmetric circular distribution is derived. Then an efficiency expression for a class of linear rank tests is obtained. The results are applied to the class of vonMises distributions and efficiency results for the sign test and the Wilcoxon test are calculated. 1. INTRODIUCTION Circular distributions are encountered in many areas of scientific investigation. Perhaps the most important example is the distribution of phases of periodic phenomena, for example, in biology and physics. Another area of application is the analysis of directions, for example, in earth sciences, migration, etc. Several examples are given by Batschelet (1965). Assume that Xl, ..., Xn are independent observations from a random variable, which takes its values on the unit circle. In this paper we are concerned with the problem of finding nonparametric tests for the hypothesis that the underlying distribution is symmetric with respect to the horizontal axis against the alternative of a displaced or rotated centre of symmetry. We proceed as follows: First we obtain a locally most powerful rank test against rotation alternatives, ? 2. After studying the distribution of a class of related test statistics under the hypothesis as well as under alternatives, ? 3, we derive the efficiency of such test sequences with respect to parametric competitors and we show that a fully efficient nonparametric test exists, ? 4. Finally, we apply some of the results to the class of von Mises distributions, ? 5.