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arXiv · 2501.10898

High-dimensional Sobolev tests on hyperspheres

Bruno Ebner·Eduardo García-Portugués·Thomas Verdebout
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Abstract

We derive the limit null distribution of the class of Sobolev tests of uniformity on the hypersphere as the dimension and the sample size diverge to infinity at arbitrary rates. The limiting non-null behavior of these tests is also explored: (i) a general consistency result for Sobolev tests in high dimensions is shown and (ii) the asymptotic power of the tests under sequences of integrated von Mises-Fisher local alternatives is obtained. The asymptotic results are applied to test for high-dimensional rotational symmetry and spherical symmetry. Numerical experiments illustrate the derived behavior of the uniformity and spherical symmetry tests under the null and under local and fixed alternatives. A real data application tests the high-dimensional normality of the cosmic microwave background.

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Bruno Ebner, Eduardo García-Portugués, Thomas Verdebout. 2025-01-18. High-dimensional Sobolev tests on hyperspheres. https://arxiv.org/abs/2501.10898

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