Open Access
October 2018 Weak convergence of a pseudo maximum likelihood estimator for the extremal index
Betina Berghaus, Axel Bücher
Ann. Statist. 46(5): 2307-2335 (October 2018). DOI: 10.1214/17-AOS1621


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 disjoint and 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.


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Betina Berghaus. Axel Bücher. "Weak convergence of a pseudo maximum likelihood estimator for the extremal index." Ann. Statist. 46 (5) 2307 - 2335, October 2018.


Received: 1 August 2016; Revised: 1 July 2017; Published: October 2018
First available in Project Euclid: 17 August 2018

zbMATH: 06964334
MathSciNet: MR3845019
Digital Object Identifier: 10.1214/17-AOS1621

Primary: 62E20 , 62G32 , 62M09
Secondary: 60G70 , 62G20

Keywords: Block maxima , Clusters of extremes , extremal index , mixing coefficients , stationary time series

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 5 • October 2018
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