Abstract
The statistical theory of extremes is extended to independent multivariate observations that are non-stationary both over time and across space. The non-stationarity over time and space is controlled via the scedasis (tail scale) in the marginal distributions. Spatial dependence stems from multivariate extreme value theory. We establish asymptotic theory for both the weighted sequential tail empirical process and the weighted tail quantile process based on all observations, taken over time and space. The results yield two statistical tests for homoscedasticity in the tail, one in space and one in time. Further, we show that the common extreme value index can be estimated via a pseudo-maximum likelihood procedure based on pooling all (non-stationary and dependent) observations. Our leading example and application is rainfall in Northern Germany.
Funding Statement
John Einmahl holds the Arie Kapteyn Chair 2019-2022 and gratefully acknowledges the corresponding research support. Ana Ferreira was partially supported by FCT-Portugal: UIDB/04621/2020, UIDB/00006/2020 and SFRH/BSAB/142912/2018; IST: P.5088. Laurens de Haan was financially supported by FCT—Fundação para a Ciência e a Tecnologia, Portugal, through the projects UIDB/00006/2020 and PTDC/MAT-STA/28649/2017. Cláudia Neves gratefully acknowledges support from EPSRC-UKRI Innovation Fellowship grant EP/S001263/1 and project FCT-UIDB/00006/2020.
Acknowledgments
We would like to thank the Editor, the Associate Editor and four anonymous referees for their detailed reviews, which helped to improve the paper substantially.
Citation
John H. J. Einmahl. Ana Ferreira. Laurens de Haan. Cláudia Neves. Chen Zhou. "Spatial dependence and space–time trend in extreme events." Ann. Statist. 50 (1) 30 - 52, February 2022. https://doi.org/10.1214/21-AOS2067
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