A0573
Title: $k$-hull depth: In between simplicial and half-space depth
Authors: Erik Mendros - Charles University (Czech Republic) [presenting]
Stanislav Nagy - Charles University (Czech Republic)
Abstract: The purpose is to introduce the $k$-hull depth, a natural generalization of the celebrated (Liu) simplicial depth for multivariate data. The $k$-hull depth of a point $x \in \mathbb{R}^d$ is defined as the probability that the convex hull of $k$ independently sampled points from a given probability distribution covers $\mathbf{x}$. For varying values of $k$, one obtains a collection of $k$-hull depth functions with different properties. Both computational and theoretical aspects of $k$-hull depths are discussed. It is shown that (i) the computation of the $k$-hull depth for any $k$ in the plane can be performed with the same complexity as for the standard simplicial depth, (ii) in a certain sense, the $k$-hull depths can be seen as ``intermediate'' between the simplicial depth and the (Tukey) half-space depth, (iii) the $k$-hull depths satisfy many plausible properties of the simplicial depth known from the literature, while (iv) the induced notion of a multivariate median based on the $k$-hull depth is for certain values of $k$ more robust than the standard simplicial median.
