On a class of Sobolev tests for symmetry, their detection thresholds, and asymptotic powers

Abstract

We consider a broad class of symmetry hypothesis testing problems that includes the problems of testing uniformity or rotational symmetry on the hypersphere $\mathcal{S}^{d-1}$, as well as the problem of testing sphericity in $\mathbb{R}^d$. For this class, we study the null and non-null behaviors of Sobolev tests, with emphasis on their consistency rates and corresponding asymptotic powers. 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 but also on the nullity of the derivatives of the angular functions characterizing the alternatives we consider; 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.