The Annals of Statistics

Inference for the limiting cluster size distribution of extreme values

Christian Y. Robert

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Any limiting point process for the time normalized exceedances of high levels by a stationary sequence is necessarily compound Poisson under appropriate long range dependence conditions. Typically exceedances appear in clusters. The underlying Poisson points represent the cluster positions and the multiplicities correspond to the cluster sizes. In the present paper we introduce estimators of the limiting cluster size probabilities, which are constructed through a recursive algorithm. We derive estimators of the extremal index which plays a key role in determining the intensity of cluster positions. We study the asymptotic properties of the estimators and investigate their finite sample behavior on simulated data.

Article information

Ann. Statist., Volume 37, Number 1 (2009), 271-310.

First available in Project Euclid: 16 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Extreme values exceedance point processes limiting cluster size distribution extremal index strictly stationary sequences


Robert, Christian Y. Inference for the limiting cluster size distribution of extreme values. Ann. Statist. 37 (2009), no. 1, 271--310. doi:10.1214/07-AOS551.

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