Parametric and nonparametric symmetries in graphical models for extremes

Abstract Coloured graphical models provide a parsimonious approach to modeling high-dimensional data by exploiting symmetries in the model parameters. In this work, we introduce the notion of colouring for extremal graphical models on generalized multivariate Pareto distributions, a natural class of limiting distributions for threshold exceedances. Thanks to a stability property of the generalized multivariate Pareto distributions, coloured extremal tree models can be defined fully nonparametrically. For more general graphs, the parametric family of Hüsler–Reiss distributions allows for two alternative approaches to coloured graphical models. We study both model classes and introduce statistical methodology for parameter estimation. It turns out that for Hüsler–Reiss tree models the different definitions of coloured graphical models coincide. In addition, we show a general parametric description of extremal conditional independence statements for Hüsler–Reiss distributions. Finally, we demonstrate that our methodology outperforms existing approaches on a real data set.