Statistics Surveys

Halfspace depth and floating body

Stanislav Nagy, Carsten Schütt, and Elisabeth M. Werner

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Abstract

Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Maximum halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the maximum depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies of measures used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth.

Article information

Source
Statist. Surv., Volume 13 (2019), 52-118.

Dates
Received: September 2018
First available in Project Euclid: 22 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ssu/1561169006

Digital Object Identifier
doi:10.1214/19-SS123

Mathematical Reviews number (MathSciNet)
MR3973130

Zentralblatt MATH identifier
07080020

Subjects
Primary: 62H05: Characterization and structure theory 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 62G35: Robustness 62H11: Directional data; spatial statistics 62H99: None of the above, but in this section

Keywords
Floating body halfspace depth measures of symmetry statistical depth Tukey depth

Rights
Creative Commons Attribution 4.0 International License.

Citation

Nagy, Stanislav; Schütt, Carsten; Werner, Elisabeth M. Halfspace depth and floating body. Statist. Surv. 13 (2019), 52--118. doi:10.1214/19-SS123. https://projecteuclid.org/euclid.ssu/1561169006


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