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.

翻译：多变量极值事件的严重程度由最大边际观测值之间的依赖性驱动。Hüsler-Reiss分布是这种极值依赖性的通用模型，通常通过变差矩阵进行参数化。为表示条件独立关系并获取稀疏参数化，我们引入了新型的Hüsler-Reiss精度矩阵。与高斯情形类似，该矩阵自然地出现在Hüsler-Reiss Pareto分布的密度表示中，并通过其零模式编码极值图结构。对于给定的任意图，我们证明了部分指定的Hüsler-Reiss变差矩阵存在唯一补全，使得其精度矩阵在图的非边位置上为零。通过使用合适的边参数估计量，我们的理论首次为图结构化的Hüsler-Reiss分布提供了一致估计量。若图结构未知，我们的方法可与近期结构学习算法结合，以联合推断图及其对应的参数矩阵。基于所提方法，我们为稀疏Hüsler-Reiss模型的统计推断设计了新工具，并在美国大规模航班延误数据及多瑙河流量数据上进行了实例验证。