Bootstrapping Extreme Value Statistics

In this thesis we investigate the use of bootstrapping schemes to estimate the variance of estimators from extreme value theory. We consider estimators for the extreme value index, the central parameter in extreme value theory, and extreme quantiles for two fundamental ap- proaches in extreme value theory. We analyse the Hill estimator for the extreme value index and the Weissman estimator for extreme quantiles from the peaks over threshold approach and the probability weighted moment estimators for the extreme value index and extreme quantiles from the block maxima approach. We find the limiting distributions of a boot- strapped sample and bootstrapped block maxima and subsequently determine the asymptotic behaviour of the bootstrapped Hill estimator and the bootstrapped probability weighted mo- ment estimators. For the latter estimators, we provide an heuristic argument to show that one may use the sample variance of bootstrapped estimators to estimate the variance of the initial estimator.