Title: Halfspace depth revisited
Authors: Stanislav Nagy - Charles University (Czech Republic)
Dusan Pokorny - Charles University (Czech Republic)
Petra Laketa - Charles University (Czech Republic) [presenting]
Abstract: A halfspace depth of a given point with respect to a probability measure is defined as the infimum of the probabilities of all the closed halfspaces that contain that point. As such, halfspace depth measures the centrality of points with respect to a given probability measure and is therefore used as a multivariate quantile. The existing literature on this interesting topic usually imposes restrictive assumptions on measure. We consider halfspace depth in a general setting, for all finite Borel measures, with the intention to collect partial results from the literature and give more general theoretical results. We specially focus on 1) when and how is it possible to reconstruct the underlying measure based on its halfspace depth function and 2) extending the so-called ray basis theorem, which gives an interesting characterization of the point with the maximal halfspace depth, called the halfspace median.