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Sobolev tests for symmetry of directional data. (English) Zbl 0551.62035
For testing a probability distribution on a compact Riemannian manifold for symmetry under the action of a given group of isometries, two classes of invariant tests are proposed and some properties noted. These tests are based on Sobolev norms and generalize E. GinĂ©’s Sobolev tests of uniformity [ibid. 3, 1243-1266 (1975; Zbl 0322.62058)].
For general compact manifolds randomization tests analogous to J. A. Wellner’s tests for the two-sample case [ibid. 7, 929-943 (1979; Zbl 0417.62044)] are suggested. For the circle, distribution-free tests of symmetry based on uniform scores are provided.

62H15 Hypothesis testing in multivariate analysis
62A01 Foundations and philosophical topics in statistics
62G10 Nonparametric hypothesis testing
53B21 Methods of local Riemannian geometry
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