Explicit bivariate simplicial depth

Erik Mendroš and Stanislav Nagy

Journal of Multivariate Analysis, 2025, vol. 205, issue C

Abstract: The simplicial depth (SD) is a celebrated tool defining elements of nonparametric and robust statistics for multivariate data. While many properties of SD are well-established, and its applications are abundant, explicit expressions for SD are known only for a handful of the simplest multivariate probability distributions. This paper deals with SD in the plane. It (i) develops a one-dimensional integral formula for SD of any properly continuous probability distribution, (ii) gives exact explicit expressions for SD of uniform distributions on (both convex and non-convex) polygons in the plane or on the boundaries of such polygons, and (iii) discusses several implications of these findings to probability and statistics: (a) An upper bound on the maximum SD in the plane, (b) an implication for a test of symmetry of a bivariate distribution, and (c) a connection of SD with the classical Sylvester problem from geometric probability.

Keywords: Simplicial depth; Statistical depth; Sylvester’s problem (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1016/j.jmva.2024.105375

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