Abstract
In this chapter, we consider an application to environmental data of a bootstrap algorithm for the adaptive estimation of the extreme value index (EVI), the primary parameter in Statistics of Extremes. The EVI estimation is performed through the recent Peaks Over Random Threshold Minimum-Variance Reduced-Bias (PORT-MVRB) estimators, which apart from scale invariant, like the classical ones, are also location invariant. These estimators depend not only on an integer tuning parameter k, the number of top order statistics involved in the estimation, but also on an extra control real parameter q, 0 ≤ q < 1, which makes them highly flexible.
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References
Araújo Santos, P., Fraga Alves, M.I., Gomes, M.I.: Peaks over random threshold methodology for tail index and quantile estimation. Revstat 4(3), 227–247 (2006)
Caeiro, F., Gomes, M.I., Henriques-Rodrigues, L.: Reduced-bias tail index estimators under a third order framework. Comm. Stat. Theor Methods 38(7), 1019–1040 (2009)
Caeiro, F., Gomes, M.I., Pestana, D.: Direct reduction of bias of the classical Hill estimator. Revstat 3(2), 111–136 (2005)
Fraga Alves, M.I., Gomes, M.I., de Haan, L.: A new class of semi-parametric estimators of the second order parameter. Portugaliae Mathematica 60(2), 194–213 (2003)
Gomes, M.I., Martins, M.J.: “Asymptotically unbiased” estimators of the tail index based on external estimation of the second order parameter. Extremes 5(1), 5–31(2002)
Gomes, M.I., Oliveira, O.: The bootstrap methodology in Statistical Extremes – choice of the optimal sample fraction. Extremes 4(4), 331–358 (2001)
Gomes, M.I., Pestana, D.: A simple second order reduced-bias’ tail index estimator. J. Stat. Comput. Simul. 77(6), 487–504 (2007)
Gomes, M.I., Pestana, D.: A sturdy reduced-bias extreme quantile (VaR) estimator. J. Am. Stat. Assoc. 102(477), 280–292 (2007)
Gomes, M.I., Fraga Alves, M.I., Araújo Santos, P.: PORT Hill and moment estimators for heavy-tailed models. Comm. Stat. Simul. Comput. 37(6), 1281–1306 (2008)
Gomes, M.I., de Haan, L., Henriques-Rodrigues, L.: Tail Index estimation for heavy-tailed models: accommodation of bias in weighted log-excesses. J. Roy. Stat. Soc. B 70(1), 31–52 (2008)
Gomes, M.I., Mendonça, S., Pestana, D.: The bootstrap methodology and adaptive reduced-bias tail index and Value-at-Risk estimation. In: Sakalauskas, L., Skiadas, C., Zavadskas, E.K. (eds.) ASMDA 2009, IMI and VGTU Editions, pp. 41–44 (2009)
Gomes, M.I., Figueiredo, F., Neves, M.M.: Adaptive estimation of heavy right tails: the bootstrap methodology in action. Extremes. DOI: 10.1007/s10687-011-0146-6 (2011)
Gomes, M.I., Henriques-Rodrigues, L., Miranda, C.: Reduced-bias location-invariant extreme value index estimation: a simulation study. Comm. Stat. Simul. Comput. 40(3), 424–447 (2011)
Hill, B.M.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975)
Resnick, S.: Heavy tail modelling and teletraffic data. Ann. Stat. 25(5), 1805–1869 (1997)
Acknowledgements
This research was partially supported by National Funds through FCT — Fundação para a Ciência e a Tecnologia, project PEst-OE/MAT/UI0006/2011, PTDC/FEDER and grant SFRH/BPD/77319/2011. We also would like to thank João Carreiras for permission to use this data set.
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Gomes, M.I., Henriques-Rodrigues, L. (2013). Adaptive PORT-MVRB Estimation of the Extreme Value Index. In: Oliveira, P., da Graça Temido, M., Henriques, C., Vichi, M. (eds) Recent Developments in Modeling and Applications in Statistics. Studies in Theoretical and Applied Statistics(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32419-2_13
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DOI: https://doi.org/10.1007/978-3-642-32419-2_13
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