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

## TL;DR

This paper develops new statistical tests for rotational symmetry on hyperspheres, extending existing models, and demonstrates their effectiveness through theoretical analysis, simulations, and an astronomy application.

## Contribution

It introduces two novel classes of hyperspherical distributions and constructs asymptotically optimal tests for rotational symmetry with both specified and unspecified axes.

## Key findings

- Tests are locally asymptotically maximin under mild conditions.
- Hybrid tests are effective against multiple alternatives.
- Monte Carlo simulations confirm finite-sample performance.

## 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.