Positive dependence is present in many real world data sets and has appealing stochastic properties that can be exploited in statistical modeling and in estimation. In particular, the notion of multivariate total positivity of order 2 ($ \mathrm{MTP}_{2} $) is a convex constraint and acts as an implicit regularizer in the Gaussian case. We study positive dependence in multivariate extremes and introduce $ \mathrm{EMTP}_{2} $, an extremal version of $ \mathrm{MTP}_{2} $. This notion turns out to appear prominently in extremes, and in fact, it is satisfied by many classical models. For a H\"usler--Reiss distribution, the analogue of a Gaussian distribution in extremes, we show that it is $ \mathrm{EMTP}_{2} $ if and only if its precision matrix is a Laplacian of a connected graph. We propose an estimator for the parameters of the H\"usler--Reiss distribution under $ \mathrm{EMTP}_{2} $ as the solution of a convex optimization problem with Laplacian constraint. We prove that this estimator is consistent and typically yields a sparse model with possibly nondecomposable extremal graphical structure. Applying our methods to a data set of Danube River flows, we illustrate this regularization and the superior performance compared to existing methods.

翻译：正相依性存在于许多现实世界的数据集中，并具有吸引人的随机性质，可在统计建模和估计中加以利用。特别地，二阶多元完全正性（$\mathrm{MTP}_{2}$）是一个凸约束，在高斯情形下起到隐式正则化器的作用。我们研究了多元极值中的正相依性，并引入了$\mathrm{EMTP}_{2}$，即$\mathrm{MTP}_{2}$的极值版本。这一概念在极值领域中显著出现，事实上，许多经典模型都满足它。对于极值中的高斯分布类比——Hüsler–Reiss分布，我们证明它是$\mathrm{EMTP}_{2}$当且仅当其精度矩阵是连通图的拉普拉斯矩阵。我们提出在$\mathrm{EMTP}_{2}$条件下Hüsler–Reiss分布参数的估计器，该估计器作为带有拉普拉斯约束的凸优化问题的解。我们证明该估计器具有一致性，并且通常会产生一个稀疏模型，该模型可能具有不可分解的极值图结构。通过将我们的方法应用于多瑙河流量数据集，我们展示了这种正则化效果以及相比现有方法的优越性能。