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Name: Frank Röttger

Talk Title: Total positivity in graphical extremes

Abstract: Engelke and Hitz (2020) introduce a general theory for conditional independence and graphical models for extremes. Similarly to Gaussian graphical models, they show that for Hüsler–Reiss graphical models, conditional independencies are encoded through a transformation of the corresponding parameter matrix Γ. Multivariate total positivity of order 2 (MTP2) is a strong form of positive dependence that induces many interesting properties in graphical modeling. A multivariate Gaussian distribution is MTP2 when its precision matrix is an M-matrix, i.e., when all the non-diagonal entries in the inverse covariance matrix are non-positive. We show that a Hüsler–Reiss distribution is MTP2 if and only if the inverse Farris transform of the parameter matrix Γ is the inverse of a diagonally dominant M-matrix. Furthermore, we prove that an extremal tree model is MTP2 if and only if its bivariate marginals are MTP2. This implies that all Hüsler–Reiss tree models are MTP2, while in comparison Gaussian tree models are only MTP2 if their covariance matrices are non-negative. Similar to Lauritzen et al. (2019) we construct a coordinate descent algorithm to find a pseudo maximum likelihood estimator under the MTP2 constraint and show that it enforces a sparse extremal graphical model. We apply this method to simulated and real data. This is joint work with Sebastian Engelke and Piotr Zwiernik.

References:

Sebastian Engelke and Adrien S. Hitz. Graphical models for extremes.Journalof the Royal Statistical Society: Series B (Statistical Methodology), 82(4):871–932, 2020.

Steffen Lauritzen, Caroline Uhler, and Piotr Zwiernik. Maximum likelihoodestimation in Gaussian models under total positivity.Ann. Statist., 47(4):1835–1863, 2019.

This talk is a contributed talk at EVA 2021.