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 . The halfspace depth of a point 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 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.
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).
"Simple halfspace depth." Electron. Commun. Probab. 27 1 - 12, 2022. https://doi.org/10.1214/22-ECP503