Multivariate extreme value theory is concerned with modeling the joint tail behavior of several random variables. Existing work mostly focuses on asymptotic dependence, where the probability of observing a large value in one of the variables is of the same order as observing a large value in all variables simultaneously. However, there is growing evidence that asymptotic independence is equally important in real world applications. Available statistical methodology in the latter setting is scarce and not well understood theoretically. We revisit nonparametric estimation and introduce rank-based M-estimators for parametric models that simultaneously work under asymptotic dependence and asymptotic independence, without requiring prior knowledge on which of the two regimes applies. Asymptotic normality of the proposed estimators is established under weak regularity conditions. We further show how bivariate estimators can be leveraged to obtain parametric estimators in spatial tail models, and again provide a thorough theoretical justification for our approach.
Michaël Lalancette was supported by the Fonds de recherche du Québec—Nature et technologies and by an Ontario Graduate Scholarship. Sebastian Engelke was supported by the Swiss National Science Foundation and the Fields Institute for Research in Mathematical Sciences. Stanislav Volgushev was supported in part by a discovery grant from NSERC of Canada and a Connaught new researcher award.
We are grateful to two reviewers for their valuable input. Their constructive comments and suggestions resulted in a substantial improvement of this paper.
"Rank-based estimation under asymptotic dependence and independence, with applications to spatial extremes." Ann. Statist. 49 (5) 2552 - 2576, October 2021. https://doi.org/10.1214/20-AOS2046