Another look at halfspace depth: flag halfspaces with applications

The halfspace depth is a well-studied tool of nonparametric statistics in multivariate spaces. We introduce a flag halfspace – an intermediary between a closed halfspace and its interior – and demonstrate that the halfspace depth can be equivalently formulated also in terms of flag halfspaces. Flag halfspaces allow us to derive theoretical results regarding the halfspace depth without the need to differentiate absolutely continuous measures from measures containing atoms, as was frequently done previously. Flag halfspaces are used to state results on the dimensionality of the halfspace median set for random samples. We prove that under mild conditions, the dimension of the sample halfspace median set of <i>d</i>-variate data cannot be <i>d</i>−1 and that for <i>d</i> = 2, the sample halfspace median set must be either a two-dimensional convex polygon or a data point.