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Title: Resampling methods for an adequate tail index estimation Authors:  Ivette Gomes - FCiencias.ID, Universidade de Lisboa and CEAUL (Portugal)
Helena Penalva - ESCE-IPS and CEAUL-Universidade de Lisboa (Portugal)
Frederico Caeiro - NOVA.ID.FCT - Universidade Nova de Lisboa (Portugal)
Manuela Neves - FCiencias.ID, Universidade de Lisboa and CEAUL (Portugal) [presenting]
Abstract: In Statistics of Extremes the extreme value index (EVI) is a central parameter. We use bootstrapping schemes to perform the choice of two nuisance parameters that appear in a recent class of EVI-estimators. Given a random sample $(X_1, \dots, X_n)$ and the associated sample of ascending order statistics $(X_{1:n}\leq \cdots\leq X_{n:n})$, the classical Hill estimator of a positive EVI is the average of the $k$ log-excesses $V_{ik}:=\ln X_{n-i+1:n}-\ln X_{n-k:n}$, $1\leq i\leq k<n$. The aforementioned class, which generalizes the Hill estimator, comes from the Lehmer mean of order $p$ of $k$ positive numbers and is defined as ${\rm L}_p(k) := 1/p \left({\sum_{i=1}^k V_{ik}^p}/{\sum_{i=1}^k V_{ik}^{p-1}}\right)$. The asymptotic behaviour of the ${\rm L}_p-$EVI-estimators has revealed very nice results in the sense of minimization of the mean square error, at optimal levels. However, for finite samples the estimates show the usual trade-off between bias and variance, depending on $k$. Besides $k$ there is also the need of the choice of $p$. Bootstrap methodology has revealed to be particularly promising in the estimation of parameters of extremes events. A bootstrap algorithm for an adaptive estimation of the tuning parameters in ${\rm L}_p(k)$ is given, allowing a reliable EVI-estimation.