Permutation Tests for Directional Data

Abstract

The two-sample problem for directional data with dimension greater than one is considered. A large family of permutation tests is proposed and studied. The statistics upon which the tests are based are related to those introduced by Gine in the context of tests for uniformity, and are defined in terms of Sobolev norms. Examples treated include the unit spheres $S^p$ and hemispheres $H^p$ for directions in $(p + 1)$-dimensional Euclidean space, and the torus $T^2 = S^1 \times S^1$ for pairs of directions in two dimensions. Computable forms of the statistics with specified consistency properties are obtained for each of these examples. Sampling from the permutation distribution is proposed as a means of implementing the tests in practice. Several tests for uniformly on the torus $T^2$ are also obtained.