Bootstrap and Other Resampling Methodologies in Statistics of Extremes

In Statistics of Extremes, the estimation of parameters of extreme or even rare events is usually done under a semi-parametric framework. The estimators are based on the largest k-ordered statistics in the sample or on the excesses over a high level u. Although showing good asymptotic properties, most of those estimators present a strong dependence on k or u with high bias when the k increases or the level u decreases. The use of resampling methodologies has revealed to be promising in the reduction of the bias and in the choice of k or u. Different approaches for resampling need to be considered depending on whether we are in an independent or in a dependent setup. A great amount of investigation has been performed for the independent situation. The main objective of this article is to use bootstrap and jackknife methods in the context of dependence to obtain more stable estimators of a parameter that appears characterizing the degree of local dependence on extremes, the so-called extremal index. A simulation study illustrates the application of those methods.