  Testing uniformity on the $p$-dimensional unit sphere is arguably the most
fundamental problem in directional statistics. In this paper, we consider this
problem in the framework of axial data, that is, under the assumption that the
$n$ observations at hand are randomly drawn from a distribution that charges
antipodal regions equally. More precisely, we focus on axial, rotationally
symmetric, alternatives and first address the problem under which the direction
$\theta$ of the corresponding symmetry axis is specified. In this setup, we
obtain Le Cam optimal tests of uniformity, that are based on the sample
covariance matrix (unlike their non-axial analogs, that are based on the sample
average). For the more important unspecified-$\theta$ problem, some classical
tests are available in the literature, but virtually nothing is known on their
non-null behavior. We therefore study the non-null behavior of the celebrated
Bingham test and of other tests that exploit the single-spiked nature of the
considered alternatives. We perform Monte Carlo exercises to investigate the
finite-sample behavior of our tests and to show their agreement with our
asymptotic results.
