Statistical Inference for Hüsler-Reiss Graphical Models Through Matrix Completions

The severity of multivariate extreme events is driven by the dependence between the largest marginal observations. The H\"usler-Reiss distribution is a versatile model for this extremal dependence, and it is usually parameterized by a variogram matrix. In order to represent conditional independence relations and obtain sparse parameterizations, we introduce the novel H\"usler-Reiss precision matrix. Similarly to the Gaussian case, this matrix appears naturally in density representations of the H\"usler-Reiss Pareto distribution and encodes the extremal graphical structure through its zero pattern. For a given, arbitrary graph we prove the existence and uniqueness of the completion of a partially specified H\"usler-Reiss variogram matrix so that its precision matrix has zeros on non-edges in the graph. Using suitable estimators for the parameters on the edges, our theory provides the first consistent estimator of graph structured H\"usler-Reiss distributions. If the graph is unknown, our method can be combined with recent structure learning algorithms to jointly infer the graph and the corresponding parameter matrix. Based on our methodology, we propose new tools for statistical inference of sparse H\"usler-Reiss models and illustrate them on large flight delay data in the U.S., as well as Danube river flow data.