## The Annals of Statistics

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

#### 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 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

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

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

https://projecteuclid.org/euclid.aos/1534492837

Digital Object Identifier
doi:10.1214/17-AOS1621

Mathematical Reviews number (MathSciNet)
MR3845019

Zentralblatt MATH identifier
06964334

#### Citation

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. https://projecteuclid.org/euclid.aos/1534492837

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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.