Estimating the extremal index through local dependence

Abstract

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.