Bootstrapping tail statistics: Tail quantile process, Hill estimator, and confidence intervals for highquantiles of heavy tailed distributions

In risk management areas such as reinsurance, the need often arises to construct a confidence interval for a quantile in the tail of the distribution. While different methods are available for this purpose, doubts have been raised about the validity of full-sample bootstrap. In this paper, we first obtain some general results on the validity of fullsample bootstrap for the tail quantile process. This opens the possibility of developing bootstrap methods based on tail statistics. Second, we develop a bootstrap method for constructing confidence intervals for high-quantiles of heavy-tailed distributions and show that it is consistent. In our simulation study, the bootstrap method for constructing confidence intervals for high quantiles performed overall better than the data tilting method, but none was uniformly the best; the data tilting method appears to be currently the preferred choice. Since the two methods are based on quite different approaches, we recommend that both methods be used side by side in applications.