Modelling multivariate extreme value distributions via Markov trees

Multivariate extreme value distributions are a common choice for modelling mul- tivariate extremes. In high dimensions, however, the construction of flexible and par- simonious models is challenging. We propose to combine bivariate extreme value dis- tributions into a Markov random field with respect to a tree. Although in general not an extreme value distribution itself, this Markov tree is attracted by a multivari- ate extreme value distribution. The latter serves as a tree-based approximation to an unknown extreme value distribution with the given bivariate distributions as margins. Given data, we learn an appropriate tree structure by Prim’s algorithm with estimated pairwise upper tail dependence coefficients or Kendall’s tau values as edge weights. The distributions of pairs of connected variables can be fitted in various ways. The resulting tree-structured extreme value distribution allows for inference on rare event probabili- ties, as illustrated on river discharge data from the upper Danube basin.