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, we consider the problem of testing rotational symmetry on the hypersphere. We adopt a semiparametric approach 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 and are 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, both for specified and unspecified symmetry axis. The tests, aimed to detect location-like and scatter-like alternatives, are combined into convenient hybrid tests that are consistent against both alternatives. We perform Monte Carlo experiments that illustrate the finite-sample performances of the proposed tests and their agreement with the asymptotic results. 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 data application.

Recent advances in directional statistics

“…As simulation from it and the computation of its pdf are far quicker than for the Kent model, this model is a particularly appealing alternative when the use of computer intensive methods is being contemplated. Other Kent-like alternatives with the advantages of the ESAG model are the scaled vMF family of Scealy and Wood (2019), which has an additional parameter controlling tail-weight, and the tangent models of García-Portugués et al (2020b).…”

Preliminary test estimation in uniformly locally asymptotically normal models

“…An extensive list of models do satisfy Assumption (A). This list includes hidden Markov models (Bickel & Ritov, 1996), quantum mechanics models (Kahn & Guta, 2009, Guta & Kiukas, 2015, time series models (Drost et al, 1997;Francq & Zakoian, 2013;Hallin et al, 1999), elliptical models (Hallin & Paindaveine, 2006, Hallin et al, 2010, multisample elliptical models (Hallin & Paindaveine, 2008;Hallin et al, 2013Hallin et al, , 2014, models for directional data (Garcia-Portugues et al, 2020;Ley et al, 2013), to mention only a few.…”

Testing axial symmetry by means of directional regression quantiles

“…Some tests of the latter hypotheses have already been discussed in the non-parametric statistical literature; see, e.g., [10] and [18] for bivariate tests and [23] for a multivariate test of conditional symmetry. Related symmetries have also been studied for directional data [4].…”

“…For example, the applications might result from the law of reflection, the axial rotation of heavenly bodies, the natural axial symmetry of simple living organisms, the axial symmetry of electrostatic potential of symmetric molecules, or from the radars rotating around an axis. The need for testing axial symmetry might also arise in the same situations when rotational symmetry is investigated for directional data; see [4] and the references therein. Furthermore, the conditional axial symmetry is also closely linked to the directional predictability of linear models with vector responses.…”