We introduce the extremal range, a local statistic for studying the spatial extent of extreme events in random fields on $\mathbb{R}^2$. 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 $s$ to a location where there is a non-exceedance. We leverage tools from excursion-set theory to study distributional properties of the extremal range, propose parametric models and predict the median extremal range at extreme threshold levels. 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 bivariate tail dependence coefficient and the extremal index of time series in classical Extreme-Value Theory. Consistent estimation of the distribution function of the extremal range for stationary random fields is proven. For non-stationary random fields, we implement generalized additive median regression to predict extremal-range maps at very high threshold levels. An application to two large daily temperature datasets, namely reanalyses and climate-model simulations for France, highlights decreasing extremal dependence for increasing threshold levels and reveals strong differences in joint tail decay rates between reanalyses and simulations.

翻译：本文引入极值范围（extremal range）这一局部统计量，用于研究$\mathbb{R}^2$上随机场极端事件的空间范围。在给定位置$s$处超出高阈值的条件下，$s$处的极值范围定义为从$s$到最近非超出位置的最小距离，该变量为随机变量。我们利用穿越集理论（excursion-set theory）的工具研究极值范围的分布性质，提出参数模型，并预测极值阈值水平下的中位数极值范围。极值范围刻画了条件极端事件的空间范围随阈值升高而缩放的速率，并将其分布性质与经典极值理论（Extreme-Value Theory）中的二元尾部相关系数及时间序列的极值指数相关联。我们证明了平稳随机场极值范围分布函数的一致性估计。对于非平稳随机场，我们采用广义加性中位数回归来预测极高阈值水平下的极值范围图。将方法应用于法国两个大型日温度数据集（再分析资料和气候模式模拟）的结果表明，极值依赖性随阈值升高而减弱，并揭示了再分析资料与模拟数据在联合尾部衰减速率上的显著差异。