Stochastic ordering in multivariate extremes

TL;DR

This paper investigates stochastic orders among multivariate max-stable distributions, establishing when these orders hold for distributions and their exponent measures, and exploring their implications in parametric models like Dirichlet and Hüsler-Reiss.

Contribution

It proves the equivalence of stochastic orders between max-stable distributions and their exponent measures, and analyzes their behavior in parametric families.

Findings

Orders hold for distributions iff they hold for exponent measures.

In dimensions ≥3, the three orders are not equivalent.

Dirichlet and Hüsler-Reiss models are PQD-ordered within their parameters.

Abstract

The article considers the multivariate stochastic orders of upper orthants, lower orthants and positive quadrant dependence (PQD) among simple max-stable distributions and their exponent measures. It is shown for each order that it holds for the max-stable distribution if and only if it holds for the corresponding exponent measure. The finding is non-trivial for upper orthants (and hence PQD order). From dimension $d\geq 3$ these three orders are not equivalent and a variety of phenomena can occur. However, every simple max-stable distribution PQD-dominates the corresponding independent model and is PQD-dominated by the fully dependent model. Among parametric models the asymmetric Dirichlet family and the H\"usler-Reiss family turn out to be PQD-ordered according to the natural order within their parameter spaces. For the H\"usler-Reiss family this holds true even for the supermodular order.