On a class of Sobolev tests for symmetry of directions, their detection thresholds, and asymptotic powers
Abstract
We consider a class of symmetry hypothesis testing problems including testing isotropy on $\mathbb{R}^d$ and testing rotational symmetry on the hypersphere $\mathcal{S}^{d-1}$. 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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2021
- DOI:
- 10.48550/arXiv.2108.09874
- arXiv:
- arXiv:2108.09874
- Bibcode:
- 2021arXiv210809874G
- Keywords:
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- Mathematics - Statistics Theory;
- 62H11;
- 62G20;
- 62H15
- E-Print:
- 22 pages, 3 figures, 2 tables. Supplementary material: 11 pages