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In Statistics of Extremes, we often have to deal with the estimation of the extreme value index, a key parameter of extreme events. The adequate estimation of this parameter is of crucial importance in the estimation of other parameters of extreme events, such as an extreme quantile, a small exceedance probability, or the return period of a high level. In this paper, we first analyze a class of kernel estimators that generalize the classical Hill estimator of the extreme value index. Then, to improve the accuracy of the estimation, we also propose a new class of reduced bias kernel estimators, parameterized with a tuning parameter that allow us to change the asymptotic mean squared error. Under suitable conditions, such class of estimators is consistent and has asymptotic normal distribution with a null dominant component of asymptotic bias. As a result, we show that further bias reduction is possible with an adequate choice of the tuning parameter. Additionally, semiparametric reduced‐bias extreme quantiles estimators based on kernel estimators of the extreme value index are also put forward. Under adequate conditions on the underlying model, we establish the consistency and asymptotic normality of these extreme quantile estimators. Finally, we analyze the log‐returns of the BOVESPA stock market index, collected from 2004 to 2016.

In this paper we consider the semi-parametric estimation of extreme quantiles of a right heavy-tail model. We propose a new Log Probability Weighted Moment estimator for extreme quantiles, which is obtained from the estimators of the shape and scale parameters of the tail. Under a second-order regular variation condition on the tail, of the underlying distribution function, we deduce the non degenerate asymptotic behaviour of the estimators under study and present an asymptotic comparison at their optimal levels. In addition, the performance of the estimators is illustrated through an application to real data.

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