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

TL;DR

This paper analyzes Sobolev tests for symmetry in directional data, revealing their detection thresholds and limitations, with applications to astronomy and Monte Carlo simulations.

Contribution

It introduces new insights into the detection thresholds and blind spots of Sobolev tests for symmetry, extending understanding of their asymptotic behavior.

Findings

Sobolev tests have a detection threshold depending on test coefficients.

Tests with only odd or even rank coefficients are insensitive to certain alternatives.

Monte Carlo simulations illustrate the asymptotic properties and limitations.

Abstract

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.