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Elliptical models are the most important family of multivariate probability distributions. We explore the properties of these distributions with respect to their halfspace depth and their illumination. The densities of elliptically symmetric distributions are expressed only in terms of the depth, the illumination, and a univariate function that can be estimated from the data. These observations set the ground for robust and nonparametric inference for (nearly) elliptical models based on the use of depth and illumination.

The halfspace depth characterization conjecture states that for any two distinct (probability) measures P and Q in the d-dimensional Euclidean space, there exists a point at which the halfspace depths of P and Q differ. Until recently, it was widely believed that this conjecture holds true for all integers \(d \ge 1\). In several research papers dealing with this problem, partial positive results towards the complete characterization of measures by their depths can be found. We provide a comprehensive review of this literature, point out to certain difficulties with some of these earlier results and construct examples of distinct (probability or finite) measures whose halfspace depths coincide at all points of the sample space, for all integers \(d > 1\).

Data depth is a nonparametric tool which may serve as an extension of quantiles to general data. Any viable depth must posses the uniform strong consistency property of its sample version. In this overview, a concise summary of the available uniform consistency results for most of the depths for functional data is given. Extensions of this theory towards random surfaces, imperfectly observed, and discontinuous functional data are studied.