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October 2021 Rank-based estimation under asymptotic dependence and independence, with applications to spatial extremes
Michaël Lalancette, Sebastian Engelke, Stanislav Volgushev
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Ann. Statist. 49(5): 2552-2576 (October 2021). DOI: 10.1214/20-AOS2046

Abstract

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.

Funding Statement

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.

Acknowledgments

We are grateful to two reviewers for their valuable input. Their constructive comments and suggestions resulted in a substantial improvement of this paper.

Citation

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Michaël Lalancette. Sebastian Engelke. Stanislav Volgushev. "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

Information

Received: 1 August 2020; Revised: 1 December 2020; Published: October 2021
First available in Project Euclid: 12 November 2021

Digital Object Identifier: 10.1214/20-AOS2046

Subjects:
Primary: 62G32
Secondary: 62F12 , 62G20 , 62G30

Keywords: Asymptotic independence , inverted max-stable distribution , M-estimation , multivariate extremes , spatial process

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 5 • October 2021
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