2022/08/08 by Petra Laketa, Laketa, Petra, Stanislav Nagy +1
Environmental Science · Mathematics · #Advanced Statistical Methods and Models #FOS: Mathematics #Soil Geostatistics and Mapping #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2208.03959
openalex publication_date 2022/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The halfspace depth of a d-dimensional point x with respect to a finite (or probability) Borel measure μ in ℝd is defined as the infimum of the μ-masses of all closed halfspaces containing x. A natural question is whether the halfspace depth, as a function of x ∈ ℝd, determines the measure μ completely. In general, it turns out that this is not the case, and it is possible for two different measures to have the same halfspace depth function everywhere in ℝd. In this paper we show that despite this negative result, one can still obtain a substantial amount of information on the support and the location of the mass of μ from its halfspace depth. We illustrate our partial reconstruction procedure in an example of a non-trivial bivariate probability distribution whose atomic part is determined successfully from its halfspace depth.