Abstract
The tail empirical process (TEP) generated by an i.i.d. sequence of regularly varying random variables is key to investigating the behaviour of extreme value statistics such as the Hill and harmonic moment estimators of the tail index. The main contribution of the paper is to prove that Efron’s bootstrap produces versions of the estimators that exhibit the same asymptotic behaviour, including possible bias. In addition, the bootstrap provides new estimators of the tail index based on variability. Further, the asymptotic behaviour of the bootstrap variance estimators is shown to be unaffected by bias.
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Data Availability
Code for all of the simulations in this article is available at: https://mysite.science.uottawa.ca/rkulik/research.html
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28 November 2022
A Correction to this paper has been published: https://doi.org/10.1007/s10687-022-00455-5
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Acknowledgements
The authors thank two anonymous referees for their helpful suggestions and careful reading of the paper. We gratefully acknowledge that the idea of investigating the effects of asymptotic bias was due to one of the referees, who suggested assumption (11) and provided the statement of Lemma 8.6 and an outline of its proof.
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Research of Gail Ivanoff and Rafal Kulik was supported by grants from the Natural Sciences and Engineering Research Council of Canada
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Ivanoff , B.G., Kulik, R. & Loukrati, H. Integral Functionals and the Bootstrap for the Tail Empirical Process. Extremes 26, 1–41 (2023). https://doi.org/10.1007/s10687-022-00445-7
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DOI: https://doi.org/10.1007/s10687-022-00445-7
Keywords
- Tail empirical process
- Regular variation
- Tail index
- Hill estimator
- Harmonic moment estimator
- Bootstrap
- Bootstrap variance estimator