Probabilities of concurrent extremes

TL;DR

This paper introduces the concept of extremal concurrence to analyze the simultaneous occurrence of extreme events across multiple locations, providing new theoretical tools and estimators for understanding spatial extreme patterns.

Contribution

It develops the notion of extremal concurrence probability, derives explicit formulas for max-stable models, and links pairwise concurrence to Kendall's tau, advancing spatial extremes analysis.

Findings

Extremal concurrence probability converges to an asymptotic measure.

Pairwise extremal concurrence equals Kendall's tau.

Application to US temperature extremes reveals meaningful concurrence patterns.

Abstract

The statistical modelling of spatial extremes has recently made major advances. Much of its focus so far has been on the modelling of the magnitudes of extreme events but little attention has been paid on the timing of extremes. To address this gap, this paper introduces the notion of extremal concurrence. Suppose that one measures precipitation at several synoptic stations over multiple days. We say that extremes are concurrent if the maximum precipitation over time at each station is achieved simultaneously, e.g., on a single day. Under general conditions, we show that the finite sample concurrence probability converges to an asymptotic quantity, deemed extremal concurrence probability. Using Palm calculus, we establish general expressions for the extremal concurrence probability through the max-stable process emerging in the limit of the componentwise maxima of the sample. Explicit forms of the extremal concurrence probabilities are obtained for various max-stable models and several estimators are introduced. In particular, we prove that the pairwise extremal concurrence probability for max-stable vectors is precisely equal to the Kendall's $\tau$. The estimators are evaluated by using simulations and applied to study the concurrence patterns of temperature extremes in the United States. The results demonstrate that concurrence probability can provide a powerful new perspective and tools for the analysis of the spatial structure and impact of extremes.