Data depth, random simplices and multivariate dispersion

Depth functions give information not only on the location but also on the dispersion of probability distributions. The Lebesgue integral of Liu’s simplicial depth function is equal to the expected volume of the random simplex whose vertices are p+1p+1 independent observations from the relevant distribution. Oja’s volume depth is the Lebesgue integral of a linear transformation of the influence function of simplicial depth. The relation of these results with dispersive orderings of distributions is discussed. Some properties of Mahalanobis’ and halfspace depth are illustrated.