"On a class of Sobolev tests for symmetry, their detection thresholds, and asymptotic powers", accepted in the Journal of the American Statistical Association

The paper “On a class of Sobolev tests for symmetry, their detection thresholds, and asymptotic powers”, coauthored by Eduardo García-Portugués, Davy Paindaveine, and Thomas Verdebout, has been accepted for publication in the renowned Journal of the American Statistical Association! The preprint can be checked on the arXiv.

This is the abstract of the paper:

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.