Halfspace depth for general measures: the ray basis theorem and its consequences

Laketa, Petra; Nagy, Stanislav

doi:10.1007/s00362-021-01259-8

Cited by 5 publications

(2 citation statements)

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“…Second, in contrast to the usual minimizing halfspaces that do not exist at certain points x ∈ R d , according to Lemma 1 there always exists a minimizing flag halfspace of any µ at any x. Besides (3), another collection of depth-generated sets of interest considered in [2,8] is…”

Section: Minimizing Halfspaces and Flag Halfspacesmentioning

confidence: 99%

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Partial reconstruction of measures from halfspace depth

Laketa¹,

Nagy²

2022

Preprint

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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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Section: Minimizing Halfspaces and Flag Halfspacesmentioning

confidence: 99%

“…let µ(F ) = α. We can write F in the form of the union {x} ∪ d i=1 relint(H i ) as in (2). Since H d ∈ H(x) is a halfspace that contains x on its boundary and x lies in the open line segment L(y, z), one of the following must hold true with j = d:…”

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confidence: 99%