On the spatial extent of extreme threshold exceedances

TL;DR

This paper introduces the extremal range, a new local statistic for analyzing the spatial extent of extreme events in random fields, linking it to established extreme-value theory concepts and providing theoretical and numerical insights.

Contribution

The paper develops the extremal range as a novel statistic for spatial extremes, deriving its distributional properties and connections to tail dependence and extremal index, with applications to common models.

Findings

Extremal range captures how the spatial extent of extremes scales with high thresholds.

Distributional properties of extremal range relate to Lipschitz-Killing curvatures and tail dependence.

Numerical studies confirm the theoretical links and estimator performance.

Abstract

We introduce the extremal range, a local statistic for studying the spatial extent of extreme events in random fields on $\mathbb{R}^d$. Conditioned on exceedance of a high threshold at a location $s$, the extremal range at $s$ is the random variable defined as the smallest distance from $s\in\mathbb{R}^d$ to a location where there is a nonexceedance. We leverage tools from excursion-set theory, such as Lipschitz- Killing curvatures, to express distributional properties of the extremal range, including asymptotics for small distances and high thresholds. The extremal range captures the rate at which the spatial extent of conditional extreme events scales for increasingly high thresholds, and we relate its distributional properties with the well-known bivariate tail dependence coefficient and the extremal index of time series in Extreme-Value Theory. We calculate theoretical extremal-range properties for commonly used models, such as Gaussian or regularly varying random fields. Numerical studies illustrate that, when the extremal range is estimated from discretized excursion sets observed on compact observation windows, the distribution of the resulting estimators appropriately reproduces the theoretically derived links with the Lipschitz- Killing curvature densities.