The Annals of Statistics

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

Betina Berghaus and Axel Bücher

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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.

Article information

Ann. Statist., Volume 46, Number 5 (2018), 2307-2335.

Received: August 2016
Revised: July 2017
First available in Project Euclid: 17 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G32: Statistics of extreme values; tail inference 62E20: Asymptotic distribution theory 62M09: Non-Markovian processes: estimation
Secondary: 60G70: Extreme value theory; extremal processes 62G20: Asymptotic properties

Clusters of extremes extremal index stationary time series mixing coefficients block maxima


Berghaus, Betina; Bücher, Axel. Weak convergence of a pseudo maximum likelihood estimator for the extremal index. Ann. Statist. 46 (2018), no. 5, 2307--2335. doi:10.1214/17-AOS1621.

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Supplemental materials

  • Supplement to: “Weak convergence of a pseudo maximum likelihood estimator for the extremal index”. The supplement contains missing proofs for the results in this paper and additional simulation results.