Bernoulli and tail-dependence compatibility

TL;DR

This paper investigates the conditions under which a matrix can represent tail-dependence coefficients of a multivariate distribution, establishing a link with Bernoulli-compatible matrices and introducing new copula models.

Contribution

It characterizes tail-dependence matrices as scaled Bernoulli-compatible matrices and introduces novel copula models for constructing such matrices.

Findings

A tail-dependence matrix with diagonal entries 1 is a scaled Bernoulli-compatible matrix.

Provides a characterization linking tail-dependence matrices to Bernoulli-compatible matrices.

Introduces new copula models for constructing tail-dependence matrices.

Abstract

The tail-dependence compatibility problem is introduced. It raises the question whether a given $d\times d$-matrix of entries in the unit interval is the matrix of pairwise tail-dependence coefficients of a $d$-dimensional random vector. The problem is studied together with Bernoulli-compatible matrices, that is, matrices which are expectations of outer products of random vectors with Bernoulli margins. We show that a square matrix with diagonal entries being 1 is a tail-dependence matrix if and only if it is a Bernoulli-compatible matrix multiplied by a constant. We introduce new copula models to construct tail-dependence matrices, including commonly used matrices in statistics.