Sobolev Tests of Goodness of Fit of Distributions on Compact Riemannian Manifolds

Classes of coordinate-invariant omnibus goodness-of-fit tests on compact Riemannian manifolds are proposed. The tests are based on Giné's Sobolev tests of uniformity. A condition for consistency is given. The tests are illustrated by an example on the rotation group SO(3). 1. Introduction. Although many tests of goodness of fit are available for distributions on the circle, comparatively little work has been done on developing general tests of goodness of fit on spheres and other sample spaces used in directional statistics. Goodness-of-fit tests for specific models include score tests for Fisher distributions within the Kent family [11], Bingham distributions within the Fisher–Bingham family [11], and for von Mises–Fisher distributions within the Fisher–Bingham family [13], as well as omnibus tests for Fisher distributions [6] and for Watson distributions [2]. An overview is given in Section 12.3 of [14]. The only general work on goodness-of-fit tests for directional distributions appears to be that of Beran [1] and of Boulerice and Ducharme [3]. Beran introduced Wald-type tests for certain nested exponential models on spheres, whereas Boulerice and Ducharme considered score tests of goodness of fit of distributions on spheres and projective spaces. Neither Beran's tests nor those of Boulerice and Ducharme are consistent against all alternatives. For continuous distributions on the real line or the circle, the probability integral transform can be used to derive a test of goodness of fit from each test of uniformity. However, if the sample space is a manifold of dimension greater than 1, then there is no unique coordinate-invariant analogue of the probability integral transform, so that it is not obvious how one can obtain tests of goodness of fit from tests of uniformity. The purpose of this paper