On the local metric property in multivariate extremes

Many multivariate data sets exhibit a form of positive dependence, which can either appear globally between all variables or only locally within particular subgroups. In models in multivariate extremes arising from threshold exceedances, a natural notion of positive dependence is the recently introduced extremal multivariate total positivity of order 2 ($ \text{EMTP}_2 $). While $ \text{EMTP}_2 $ has nice theoretical properties, it is by construction a global property and therefore not suitable for applications with only local positive dependence. We introduce extremal association as a weaker form of extremal positive dependence and show that it generalizes extremal tree models. This follows from a sufficient condition for extremal association, which for H\"usler--Reiss distributions 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 electric 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.