Estimating the extremal index through local dependence

TL;DR

This paper introduces a new method for estimating the extremal index in stationary sequences, leveraging local dependence conditions, and evaluates its performance through simulations and real data application.

Contribution

It proposes a novel estimation approach based on local dependence conditions and provides empirical diagnostics, with comprehensive simulation and real data analysis.

Findings

The new estimator performs well in simulations.

The empirical diagnostic effectively identifies local dependence conditions.

Application to financial data demonstrates practical utility.

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.