Invariant Tests for Uniformity on Compact Riemannian Manifolds Based on Sobolev Norms

Abstract

Several invariant tests for uniformity of a distribution on the circle, the sphere and the hemisphere have been proposed by Rayleigh, Watson, Bingham, Ajne, Beran and others. In this paper a class of invariant tests for uniformity on compact Riemannian manifolds containing many of the known ones is presented and studied (the asymptotic theory as well as some local optimality properties for this class of tests are given). The examples include two new tests, one for the sphere and the other for the hemisphere. Let $X$ be a compact Riemannian manifold, $\mu$ the normalized volume element (the uniform distribution of $X$) and $\nu_n$ the empirical distribution corresponding to a sequence of i.i.d. $X$-valued random variables. The statistics in which these tests are based are just convergent weighted sums of the squares of the Fourier coefficients of $\nu_n(\omega) - \mu$ with respect to any orthonormal basis of $L_2(X, \mu)$ consisting of eigenfunctions of the Laplacian. An additional condition is imposed on the weights, namely that weights corresponding to coefficients of eigenfunctions in the same eigenspace of the Laplacian be equal (this condition is essential for the invariance of the tests). These statistics are related to Sobolev norms and so, the tests are called Sobolev tests. In connection with Sobolev statistics, it is interesting to note that the Sobolev norms of index $-s, s > (\dim X)/2$, metrize the weak-star topology of $\mathscr{P}(X)$, the space of Borel probability measures on $X$. A theorem about weak convergence of empirical distributions on compact manifolds, useful in proving some of the asymptotic results for Sobolev statistics, is also included. One of the sections (Section 2) is almost entirely devoted to give a short review of the facts needed in the paper about Riemannian manifolds, the Laplacian and Sobolev spaces.