Simplicial Depth in the Plane: Exact Expressions and Implications

E. Mendroš

{}^{1}

S. Nagy

{}^{2}

${}^{1}$

Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

[erikmendros@gmail.com]
${}^{2}$

Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

[nagy@karlin.mff.cuni.cz]

1 Explicit Bivariate Simplicial Depth

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. In our contribution, we deal with SD in the plane. We start by developing a one-dimensional integral formula for SD of any properly continuous probability distribution. We then apply this formula to derive 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. Additionally, we discuss 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.