Total positivity in multivariate extremes

Positive dependence is present in many real world data sets and has appealing stochastic properties. In particular, the notion of multivariate total positivity of order 2 ($ \text{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 $ \text{EMTP}_2 $, an extremal version of $ \text{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 $ \text{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 $ \text{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 non-decomposable extremal graphical structure. At the example of two data sets, we illustrate this regularization and the superior performance compared to existing methods.