Weak convergence of a pseudo maximum likelihood estimator for the extremal index

TL;DR

This paper analyzes the asymptotic properties of disjoint and sliding blocks estimators for the extremal index in stationary time series, showing the sliding blocks estimator's superior performance through theoretical and simulation results.

Contribution

It introduces a pseudo maximum likelihood estimator for the extremal index with only one parameter, deriving its asymptotic properties and demonstrating its effectiveness.

Findings

Sliding blocks estimator outperforms other blocks estimators.

Estimator is competitive with runs- and inter-exceedance estimators.

Methods are successfully applied to financial time series.

Abstract

The extremes of a stationary time series typically occur in clusters. A primary measure for this phenomenon is the extremal index, representing the reciprocal of the expected cluster size. Both a disjoint and a sliding blocks estimator for the extremal index are analyzed in detail. In contrast to many competitors, the estimators only depend on the choice of one parameter sequence. We derive an asymptotic expansion, prove asymptotic normality and show consistency of an estimator for the asymptotic variance. Explicit calculations in certain models and a finite-sample Monte Carlo simulation study reveal that the sliding blocks estimator outperforms other blocks estimators, and that it is competitive to runs- and inter-exceedance estimators in various models. The methods are applied to a variety of financial time series.