Probabilities of Concurrent Extremes

The statistical modeling of spatial extremes has been an active area of recent research with a growing domain of applications. Much of the existing methodology, however, focuses on the magnitudes of extreme events rather than on their timing. To address this gap, this article investigates the notion of extremal concurrence. Suppose that daily temperatures are measured at several synoptic stations. We say that extremes are concurrent if record maximum temperatures occur simultaneously, that is, on the same day for all stations. It is important to be able to understand, quantify, and model extremal concurrence. 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 component-wise 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 τ. The estimators are evaluated from simulations and applied to study temperature extremes in the United States. Results demonstrate that concurrence probability can be used to study, for example, the effect of global climate phenomena such as the El Niño Southern Oscillation (ENSO) or global warming on the spatial structure and areal impact of extremes.