Methods and Algorithms to Test the Hausdorff and Simplex Dispersion Orders with an R Package

Abstract

Stochastic orders aim to order probability distributions in accordance with an appropriate criterion. Dispersion orderings are particular cases of stochastic orderings. Essentially, given two random vectors, a dispersion ordering attempts to determine which vector induces a more dispersive probability distribution. The Hausdorff and simplex dispersion orderings are two particular cases of such a kind of orders. Although they satisfy suitable properties from a theoretical point of view, the application to real problems is very complex since the study of such orders implies to determine sample values of Hausdorff distances between random convex hulls. The paper proposes two exact algorithms to test the Hausdorff and simplex dispersion orderings. A software implementation using R is provided and evaluated using a simulation study. An ophthalmological application concerned with the diabetes evaluation using the mean calibers of arteries and veins in fundus images is considered. The Hausdorff and simplex dispersion orderings are applied to the study of the effects produced by diabetes in the retinal vessels. The possible differences in dispersion that could exist between the groups defined using some categorical covariables are tested. The comparison between homogeneous groups will produce accurate results in medical research.