Clustering of extreme events can have profound and detrimental societal consequences. The extremal index, a number in the unit interval, is a key parameter in modelling the clustering of extremes. The study of extremal index often assumes a local dependence condition known as the $D^{(d)}(u_n)$ condition. In this paper, we develop a hypothesis test for $D^{(d)}(u_n)$ condition based on asymptotic results. We develop an estimator for the extremal index by leveraging the inference procedure based on the $D^{(d)}(u_n)$ condition, and we establish the asymptotic normality of this estimator. The finite sample performances of the hypothesis test and the estimation are examined in a simulation study, where we consider both models that satisfies the $D^{(d)}(u_n)$ condition and models that violate this condition. In a simple case study, our statistical procedure shows that daily temperature in summer shares a common clustering structure of extreme values based on the data observed in three weather stations in the Netherlands, Belgium and Spain.

翻译：极端事件的聚集可能对社会造成深远且有害的影响。极值指数是单位区间内的一个数值，是建模极端值聚集的关键参数。极值指数的研究通常假设一个称为$D^{(d)}(u_n)$条件的局部依赖条件。在本文中，我们基于渐近结果开发了针对$D^{(d)}(u_n)$条件的假设检验。我们通过利用基于$D^{(d)}(u_n)$条件的推断过程，提出了一种极值指数的估计方法，并建立了该估计量的渐近正态性。通过模拟研究，我们考察了假设检验和估计的有限样本表现，其中考虑了满足$D^{(d)}(u_n)$条件的模型以及违反该条件的模型。在一个简单的案例研究中，我们的统计方法表明，根据荷兰、比利时和西班牙三个气象站的观测数据，夏季日温度共享一种共同的极端值聚集结构。