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Estimation of the Weibull Tail Coefficient Through the Power Mean-of-Order-p

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Recent Developments in Statistics and Data Science (SPE 2021)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 398))

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Abstract

The Weibull tail coefficient (WTC) is the parameter \(\theta \) in a right-tail function of the type \(\overline{F}:=1-F\), such that \(H:=-\ln \overline{F}\) is a regularly varying function at infinity with an index of regular variation equal to \(\theta \in \mathbb {R}^{+}\). In a context of extreme value theory for maxima, it is possible to prove that we have an extreme value index (EVI) \(\xi =0\), but usually a very slow rate of convergence. Most of the recent WTC-estimators are proportional to the class of Hill EVI-estimators, the average of the log-excesses associated with the k upper order statistics, \(1\le k<n\). The interesting performance of EVI-estimators based on generalized means leads us to base the WTC-estimation on the power mean-of-order-p (MO\(_p\)) EVI-estimators. Consistency of the WTC-estimators is discussed and their performance, for finite samples, is illustrated through a small-scale Monte Carlo simulation study.

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Acknowledgements

The research was partially supported by National Funds through FCT—Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through projects UIDB/00297/2020 (CMA/UNL), UIDB/00006/2020 (CEA/UL) and UIDB/04674/2020 (CIMA).

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Authors and Affiliations

  1. NOVA School of Science and Technology (FCT NOVA) and CMA, NOVA University of Lisbon, Campus de Caparica, Lisbon, Portugal

    Frederico Caeiro

  2. Faculty of Science of Lisbon (FCUL/DEIO) and CEAUL, University of Lisbon, Cidade Universitária, Campo Grande, Lisbon, Portugal

    M. Ivette Gomes

  3. School of Science and Technology (ECT-UE) and CIMA, University of Évora, Évora, Portugal

    Lígia Henriques-Rodrigues

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  1. Frederico Caeiro
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  2. M. Ivette Gomes
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  3. Lígia Henriques-Rodrigues
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Correspondence to Frederico Caeiro .

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Editors and Affiliations

  1. NOVA School of Science and Technology, Caparica, Portugal

    Regina Bispo

  2. University of Évora, Évora, Portugal

    Lígia Henriques-Rodrigues

  3. University of Évora, Évora, Portugal

    Russell Alpizar-Jara

  4. University of Edinburgh, Edinburgh, UK

    Miguel de Carvalho

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Caeiro, F., Gomes, M.I., Henriques-Rodrigues, L. (2022). Estimation of the Weibull Tail Coefficient Through the Power Mean-of-Order-p. In: Bispo, R., Henriques-Rodrigues, L., Alpizar-Jara, R., de Carvalho, M. (eds) Recent Developments in Statistics and Data Science. SPE 2021. Springer Proceedings in Mathematics & Statistics, vol 398. Springer, Cham. https://doi.org/10.1007/978-3-031-12766-3_4

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