Supermodular ordering of binomial, Poisson and Gaussian random vectors by tree-based correlations

TL;DR

This paper introduces a tree-based method to compare binomial, Poisson, and Gaussian random vectors with given covariances, using supermodular ordering and dependence structures.

Contribution

It presents a novel tree-based dependence construction for these vectors, linking supermodular ordering to covariance matrix ordering, and employs M"obius inversion techniques.

Findings

Characterizes supermodular ordering via covariance matrices.

Provides a dependence structure construction using binary trees.

Connects Poisson and Gaussian vectors through approximation methods.

Abstract

We construct a tree-based dependence structure for the representation of binomial, Poisson and Gaussian random vectors having a given covariance matrix, using sums of independent random variables. This construction allows us to characterize the supermodular ordering of such random vectors via the componentwise ordering of their covariance matrices. Our method relies on the representation of dependent components using binary trees on the discrete $d$-dimensional hypercube $C_d$, and on M\"obius inversion techniques. In the case of Poisson random vectors this approach involves L\'evy measures on $C_d$, and it is consistent with the approximation of Poisson and multivariate Gaussian random vectors by binomial vectors.