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On a class of Sobolev tests for symmetry of directions, their detection thresholds, and asymptotic powers
Eduardo García-Portugués, Davy Paindaveine, Thomas Verdebout
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We consider a class of symmetry hypothesis testing problems including testing isotropy on ℝᵈ and testing rotational symmetry on the hypersphere 𝒮ᵈ⁻¹. For this class, we study the null and non-null behaviors of Sobolev tests, with emphasis on their consistency rates. Our main results show that: (i) Sobolev tests exhibit a detection threshold (see Bhattacharya, 2019, 2020) that does not only depend on the coefficients defining these tests; and (ii) tests with non-zero coefficients at odd (respectively, even) ranks only are blind to alternatives with angular functions whose $k$th-order derivatives at zero vanish for any k odd (even). Our non-standard asymptotic results are illustrated with Monte Carlo exercises. A case study in astronomy applies the testing toolbox to evaluate the symmetry of orbits of long- and short-period comets.
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