Clustering of extremes has a large societal impact. The extremal index, a number in the unit interval, is a key parameter in modelling the clustering of extremes. We build a connection between the extremal index and the stable tail dependence function, which enables us to compute the value of extremal indices for some time series models. We also construct a nonparametric estimator of the extremal index and an estimation procedure to verify $D^{(d)}(u_n)$ condition, a local dependence condition often assumed for studying the extremal index.

We prove that the estimator is asymptotically normal. The simulation study which compares our estimator to two existing methods shows that our method has better finite sample properties. We apply our method to estimate the expected durations of heatwaves in the Netherlands and in Greece.