Weak convergence of a pseudo maximum likelihood estimator for the extremal index
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Date
2016
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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 is outperforming 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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Keywords
clusters of extremes, block maxima, mixing coefficients, stationary time series, extremal index