A new semi-parametric family of estimators for the second order parameter

International audienceAn important parameter in extreme value theory is the extreme value index $\gamma$. 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). The most known estimator was proposed by [2]. It depends on the $k$ largest observations of the underlying sample. The bias of the tail index estimator is controlled by the second order parameter $\rho$. In order to reduce the bias of $\gamma$'s estimators or to select the best number $k$ of observations to use, the knowledge of $\rho$ is essential. Some estimators of $\rho$ can be found in the literature, see for example [1, 2, 3]. We propose a semiparametric family of estimators for $\rho$ that encompasses the three previously mentioned estimators. The asymptotic normality of these estimators is then proved in an uni fied way. New estimators of $\rho$ are also introduced