Statistical depth: Geometry of multivariate quantiles

Abstract

Statistical depth is a non-parametric tool applicable to multivariate and non-Euclidean data. Its goal is to reasonably generalize quantiles to multivariate and more exotic datasets. The first depth was proposed in statistics in 1975; rigorous investigation of depths started in the 1990s, and still, an abundance of open problems stimulates research in the area. We discuss two seminal depths: (i) the halfspace depth (Tukey, 1975) and (ii) the simplicial depth (Liu, 1988). We unveil surprising links between these depths and well-studied concepts from geometry and discrete mathematics. Using these relations, we partially resolve several open problems, in particular, the 30-year-old characterization conjecture, asking whether two different distributions can correspond to the same halfspace depth.

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