Abstract
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 () is a convex constraint and acts as an implicit regularizer in the Gaussian case. We study positive dependence in multivariate extremes and introduce , an extremal version of . 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 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 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.
Funding Statement
Sebastian Engelke and Frank Röttger were supported by the Swiss National Science Foundation (Grant 186858). Piotr Zwiernik acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), grant RGPIN-2023-03481.
Acknowledgments
The authors would like to thank the two anonymous referees, an Associate Editor and the Editor for their constructive comments that strongly improved the quality of this paper.
Citation
Frank Röttger. Sebastian Engelke. Piotr Zwiernik. "Total positivity in multivariate extremes." Ann. Statist. 51 (3) 962 - 1004, June 2023. https://doi.org/10.1214/23-AOS2272
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