Open Access
2022 Simple halfspace depth
Petra Laketa, Dušan Pokorný, Stanislav Nagy
Author Affiliations +
Electron. Commun. Probab. 27: 1-12 (2022). DOI: 10.1214/22-ECP503


The halfspace depth is a prominent tool of nonparametric inference for multivariate data. We consider it in the general context of finite Borel measures μ on Rd. The halfspace depth of a point xRd is defined as the infimum of the μ-masses of halfspaces that contain x. We say that a measure μ has a simple (halfspace) depth if the set of all attained halfspace depth values of μ on Rd is finite. We give a complete description of measures with simple depths by showing that the halfspace depth of μ is simple if and only if μ is atomic with finitely many atoms. This result completely resolves the halfspace depth characterization problem for the particular situation of simple halfspace depths and datasets. We also discuss the cardinality of the set of the attained halfspace depth values.

Funding Statement

P. Laketa was supported by the OP RDE project “International mobility of research, technical and administrative staff at the Charles University”, grant CZ.02.2.69/0.0/0.0/18_053/0016976. The work of S. Nagy was supported by Czech Science Foundation (EXPRO project n. 19-28231X).


Download Citation

Petra Laketa. Dušan Pokorný. Stanislav Nagy. "Simple halfspace depth." Electron. Commun. Probab. 27 1 - 12, 2022.


Received: 10 April 2022; Accepted: 27 November 2022; Published: 2022
First available in Project Euclid: 14 December 2022

MathSciNet: MR4529627
zbMATH: 07644377
Digital Object Identifier: 10.1214/22-ECP503

Primary: 62G35 , 62H05

Keywords: characterization , flag halfspace , halfspace depth , Statistical depth , Tukey depth

Back to Top