Total positivity in multivariate extremes
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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.
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