  The angular halfspace depth (ahD) is a natural modification of the celebrated
halfspace (or Tukey) depth to the setup of directional data. It allows us to
define elements of nonparametric inference, such as the median, the
inter-quantile regions, or the rank statistics, for datasets supported in the
unit sphere. Despite being introduced in 1987, ahD has never received ample
recognition in the literature, mainly due to the lack of efficient algorithms
for its computation. With the recent progress on the computational front, ahD
however exhibits the potential for developing viable nonparametric statistics
techniques for directional datasets. In this paper, we thoroughly treat the
theoretical properties of ahD. We show that similarly to the classical
halfspace depth for multivariate data, also ahD satisfies many desirable
properties of a statistical depth function. Further, we derive uniform
continuity/consistency results for the associated set of directional medians,
and the central regions of ahD, the latter representing a depth-based analogue
of the quantiles for directional data.
