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.

翻译：本文考虑一类对称性假设检验问题，包括$\mathbb{R}^d$上的各向同性检验和超球面$\mathcal{S}^{d-1}$上的旋转对称性检验。针对此类问题，我们研究Sobolev检验在原假设和备择假设下的行为，重点关注其一致性速率。主要结果表明：（i）Sobolev检验存在一个检测阈值（参见 Bhattacharya, 2019, 2020），该阈值不仅取决于定义这些检验的系数；（ii）仅在奇数（或偶数）秩上具有非零系数的检验，对于角函数在零点处所有奇数（或偶数）阶导数均为零的备择假设完全失效。我们采用蒙特卡洛实验验证了这些非标准渐近结果。一项天文学案例研究应用该检验工具箱评估了长周期与短周期彗星轨道对称性。