A simple class of reduced bias kernel estimators of the extreme value index

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 the 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, semi‐parametric 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.