On the local metric property in multivariate extremes
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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 (EMTP_2). While 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 Hüsler–Reiss distributions permits a parametric description that we call the metric property. As the parameter of a Hüsler–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 Hüsler–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.
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