Tail inference using extreme U-statistics
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Extreme U-statistics arise when the kernel of a U-statistic has a high degree but depends only on its arguments through a small number of top order statistics. As the kernel degree of the U-statistic grows to infinity with the sample size, estimators built out of such statistics form an intermediate family in between those constructed in the block maxima and peaks-over-threshold frameworks in extreme value analysis. The asymptotic normality of extreme U-statistics based on location-scale invariant kernels is established. Although the asymptotic variance corresponds with the one of the Hájek projection, the proof goes beyond considering the first term in Hoeffding's variance decomposition; instead, a growing number of terms needs to be incorporated in the proof. To show the usefulness of extreme U-statistics, we propose a kernel depending on the three highest order statistics leading to an unbiased estimator of the shape parameter of the generalized Pareto distribution. When applied to samples in the max-domain of attraction of an extreme value distribution, the extreme U-statistic based on this kernel produces a location-scale invariant estimator of the extreme value index which is asymptotically normal and whose finite-sample performance is competitive with that of the pseudo-maximum likelihood estimator.
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