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)(unβ) condition, is often assumed.
In this paper, we develop a hypothesis test for D(d)(unβ) condition based
on asymptotic results. Further we construct an estimator of the extremal index
making use of the inference procedure on D(d)(unβ) 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)(unβ) 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