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.

翻译：角半空间深度（ahD）是著名的半空间（或Tukey）深度在方向数据情境下的自然修正。它使我们能够为支持在单位球面上的数据集定义非参数推断的要素，例如中位数、分位数间区域或秩统计量。尽管早在1987年就被提出，但ahD在文献中从未得到充分认可，主要原因是缺乏高效的计算算法。然而，随着计算领域的最新进展，ahD展现出为方向数据集开发可行的非参数统计技术的潜力。本文全面研究了ahD的理论性质。我们证明，与经典的多变量数据半空间深度类似，ahD也满足统计深度函数的许多理想性质。此外，我们推导了相关方向中位数集以及ahD中心区域的均匀连续性/一致性结果，后者代表了方向数据基于深度的分位数对应物。