Simplicial depth: Characterization and reconstruction

Statistical depth functions have been designed with the intention of extending nonparametric inference toward multivariate setups. As such, the depths should serve as multivariate analogues of the quantile functions known from the analysis of real‐valued data. The so‐called characterization and reconstruction questions are among the fundamental open problems of the contemporary depth research. Roughly speaking, they ask: (a) Is it is possible that two different datasets, or more generally, two different probability distributions, correspond to identical depths, or does the depth function uniquely characterize the underlying distribution? (b) Knowing a depth function, can we reconstruct the corresponding distribution? For any given depth to constitute a fully‐fledged alternative to the quantile function, the depth must characterize wide classes of probability measures, and these measures must be simple to recover from their depths. We investigate these characterization/reconstruction questions for the classical simplicial depth for multivariate data. We show that, under mild conditions, datasets (represented by measures putting equal mass 1/n$$ 1/n $$ to each datum in a dataset of size n$$ n $$ ) and atomic measures are characterized by, and can be easily reconstructed from, their simplicial depth.