Some Simple U-Statistic Tests for Uniformity on Certain Homogeneous Spaces

We develop some examples of degenerate U-statistic tests for uniformity in certain compact topological groups and homogeneous spaces, namely, the orthogonal group and the Grassmann manifold. These tests have the attractive properties that they involve simple matrix calculations and their asymptotic null distributions are easy to describe, in that the eigenspace decompositions that characterize these distributions are quite elementary. The tests are shown to be locally most powerful against suitable families of alternatives.