On optimal tests for rotational symmetry against new classes of hyperspherical distributions

Abstract

Motivated by the central role played by rotationally symmetric distributions in directional statistics, weconsider the problem of testing rotational symmetry on the hypersphere. We adopt a semiparametricapproach and tackle problems where the location of the symmetry axis is either specified or unspecified.For each problem, we define two tests and study their asymptotic properties under very mild conditions.We introduce two new classes of directional distributions that extend the rotationally symmetric class andare of independent interest. We prove that each test is locally asymptotically maximin, in the Le Cam sense,for one kind of the alternatives given by the new classes of distributions, for both specified and unspecifiedsymmetry axis. The tests, aimed to detect location- and scatter-like alternatives, are combined into convenient hybrid tests that are consistent against both alternatives. We perform Monte Carlo experiments thatillustrate the finite-sample performances of the proposed tests and their agreement with the asymptoticresults. Finally, the practical relevance of our tests is illustrated on a real data application from astronomy.The R package rotasym implements the proposed tests and allows practitioners to reproduce the dataapplication. Supplementary materials for this article are available online.