On a class of Sobolev tests for symmetry of directions, their detection thresholds, and asymptotic powers
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866916172473040896 |
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| author | García-Portugués, Eduardo Paindaveine, Davy Verdebout, Thomas |
| author_facet | García-Portugués, Eduardo Paindaveine, Davy Verdebout, Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_09874 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On a class of Sobolev tests for symmetry of directions, their detection thresholds, and asymptotic powers García-Portugués, Eduardo Paindaveine, Davy Verdebout, Thomas Statistics Theory 62H11, 62G20, 62H15 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. |
| title | On a class of Sobolev tests for symmetry of directions, their detection thresholds, and asymptotic powers |
| topic | Statistics Theory 62H11, 62G20, 62H15 |
| url | https://arxiv.org/abs/2108.09874 |