  The extremal index is an important parameter in the characterization of
extreme values of a stationary sequence. Our new estimation approach for this
parameter is based on the extremal behavior under the local dependence
condition D$^{(k)}$($u_n$). We compare a process satisfying one of this
hierarchy of increasingly weaker local mixing conditions with a process of
cycles satisfying the D$^{(2)}$($u_n$) condition. We also analyze local
dependence within moving maxima processes and derive a necessary and sufficient
condition for D$^{(k)}$($u_n$). In order to evaluate the performance of the
proposed estimators, we apply an empirical diagnostic for local dependence
conditions, we conduct a simulation study and compare with existing methods. An
application to a financial time series is also presented.
