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 o fLiu's simplicial depth function is equal to the expected volume of the random simplex whose vertices are p + 1 independent observations from there levant distribution. Oja'svolume depthisthe Lebesgueintegral 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.