k-hull depth: in between simplicial and halfspace depth

This paper introduces the k-hull depth, a generalized version of the celebrated (Liu) simplicial depth for multivariate data. The k-hull depth of a point $$\varvec{x}\in \mathbb R^d$$ x ∈ R d is defined as the probability that the convex hull of k independently sampled points from a given probability distribution covers $$\varvec{x}$$ x . For varying values of k, one obtains a collection of k-hull depth functions with different properties. We consider both the computation and the theoretical properties of k-hull depths. We show 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) halfspace 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. The practical relevance of considering k-hull depths is explored through simulations.