Tail inference using extreme U-statistics

Jochem Oorschot, Johan Segers () and Chen Zhou
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Jochem Oorschot: Erasmus University Rotterdam
Johan Segers: Université catholique de Louvain, LIDAM/ISBA, Belgium
Chen Zhou: Erasmus University Rotterdam

No 2023006, LIDAM Reprints ISBA from Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA)

Abstract: 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 coincides with the one of the Hájek projection, the proof goes beyond considering the first term in Hoeffding’s variance decomposition. We propose a kernel depending on the three highest order statistics leading to a location-scale invariant estimator of the extreme value index resembling the Pickands estimator. This extreme Pickands U-estimator is asymptotically normal and its finite-sample performance is competitive with that of the pseudo-maximum likelihood estimator.

Keywords: Extreme value index; generalized Pareto distribution; Hájek projection; U-statistic (search for similar items in EconPapers)
Pages: 47
Date: 2023-04-13
Note: In: Electronic Journal of Statistics, 2023, vol. 17(1), p. 1113-1159
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Persistent link: https://EconPapers.repec.org/RePEc:aiz:louvar:2023006

DOI: 10.1214/23-EJS2129

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