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On the estimation of the second order parameter for heavy-tailed distributions

El Hadji Deme 1, 2 Laurent Gardes 3 Stéphane Girard 2
2 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology
Abstract : The extreme-value index is an important parameter in extreme-value theory since it controls the fi rst order behavior of the distribution tail. In the literature, numerous estimators of this parameter have been proposed especially in the case of heavy-tailed distributions, which is the situation considered here. Most of these estimators depend on the k largest observations of the underlying sample. Their bias is controlled by the second order parameter. In order to reduce the bias of extreme-value index estimators or to select the best number k of observations to use, the knowledge of the second order parameter is essential. In this paper, we propose a simple approach to estimate the second order parameter leading to both existing and new estimators. We establish a general result that can be used to easily prove the asymptotic normality of a large number of estimators proposed in the literature or to compare di erent estimators within a given family. Some illustrations on simulations are also provided.
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Submitted on : Friday, November 16, 2012 - 11:45:50 AM
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El Hadji Deme, Laurent Gardes, Stéphane Girard. On the estimation of the second order parameter for heavy-tailed distributions. REVSTAT - Statistical Journal, Instituto Nacional de Estatistica, 2013, 11 (3), pp.277-299. ⟨hal-00634573v4⟩

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