Simplicial depth and its median: Selected properties and limitations

Depth functions are important tools of nonparametric statistics that extend orderings, ranks, and quantiles to the setup of multivariate data. We revisit the classical definition of the simplicial depth and explore its theoretical properties when evaluated with respect to datasets or measures that do not necessarily possess a symmetric density. Recent advances from discrete geometry are used to refine the results about the robustness and continuity of the simplicial depth and its induced multivariate median. Further, we compute the exact simplicial depth in several scenarios and point out some undesirable behavior: (i) the simplicial depth does not have to be maximized at the center of symmetry of the distribution, (ii) it is not necessarily unimodal, and can possess local extremes, and (iii) the sets of the induced multivariate medians or other central regions do not have to be connected.