Total positivity in multivariate extremes
share
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 (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 EMTP_2, an extremal version of MTP_2. This notion turns out to appear prominently in extremes and, in fact, it is satisfied by many classical models. For a Hüsler–Reiss distribution, the analogue of a Gaussian distribution in extremes, we show that it is 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 Hüsler–Reiss distribution under 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.
READ FULL TEXT
