On the local metric property in multivariate extremes

TL;DR

This paper introduces extremal positive association for multivariate extremes, generalizes extremal tree models, and develops a localized estimation method for Hüsler--Reiss graphical models based on the metric property.

Contribution

It defines extremal positive association, extends extremal tree models, and proposes a localized estimation approach for Hüsler--Reiss models using the metric property.

Findings

The metric property characterizes Hüsler--Reiss distributions with a Euclidean distance matrix.

A two-step estimation procedure for locally metrical Hüsler--Reiss models is developed.

The methods are validated on simulated and real datasets.

Abstract

Many multivariate data sets exhibit a form of positive dependence, which can either appear globally between all variables or only locally within particular subgroups. A popular notion of positive dependence that allows for localized positivity is positive association. In this work we introduce the notion of extremal positive association for multivariate extremes from threshold exceedances. Via a sufficient condition for extremal association, we show that extremal association generalizes extremal tree models. For H\"usler--Reiss distributions the sufficient condition permits a parametric description that we call the metric property. As the parameter of a H\"usler--Reiss distribution is a Euclidean distance matrix, the metric property relates to research in electrical network theory and Euclidean geometry. We show that the metric property can be localized with respect to a graph and study surrogate likelihood inference. This gives rise to a two-step estimation procedure for locally metrical H\"usler--Reiss graphical models. The second step allows for a simple dual problem, which is implemented via a gradient descent algorithm. Finally, we demonstrate our results on simulated and real data.