On a Class of Sobolev Tests for Symmetry, their Detection Thresholds, and Asymptotic Powers

We consider a broad class of symmetry hypothesis testing problems that includes the problems of testing uniformity or rotational symmetry on the hypersphere Sd−1, as well as the problem of testing sphericity in Rd. 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 that depends not only 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 nonzero coefficients at odd (respectively, even) ranks only are blind to alternatives with angular functions whose kth-order derivatives at zero vanish for any k odd (even). Our nonstandard 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. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.