Statistical inference on $D^{(d)}(u_n)$ condition and estimation of the Extremal Index

Clustering of extreme events often has a destructive societal impact. The extremal index, a number in the unit interval, is a key parameter in modelling the clustering of extremes. While studying the extremal index, a local dependence condition referred to as $D^{(d)}(u_n)$ condition, is often assumed. In this paper, we develop a hypothesis test for $D^{(d)}(u_n)$ condition based on asymptotic results. Further we construct an estimator of the extremal index making use of the inference procedure on $D^{(d)}(u_n)$ condition and we prove this estimator is asymptotically normal. The finite sample performances of the hypothesis test and the estimation are examined in a simulation study, where we consider models fulfilling $D^{(d)}(u_n)$ condition as well as models that violate the 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.