Partial reconstruction of measures from halfspace depth
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The halfspace depth of a d-dimensional point x with respect to a finite (or probability) Borel measure μ in ℝ^d 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 ∈ℝ^d, 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 ℝ^d. 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.
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