Optimal Confidence Intervals for the Tail Index and High Quantiles
The aim of the paper is to obtain confidence intervals for the tail index and high quantiles taking into account the optimal rate of convergence of the estimator. The common approach to obtaining confidence intervals presented in the literature is to use the normal distribution approximation at a non-optima1 rate. Instead, we propose to use the optimal rate, but then a bias term with unknown sign has to be estimated. We provide an estimator for this sign and the full programme to obtain the optimal confidence intervals. Moreover, we demonstrate the gain in coverage, and show the relevance of these confidence intervals by calculating the reduction in capital requirements in a financia1 Value at Risk exercise. Simulation results are also presented. It is weIl known that extreme value parameter estimators which balance the asymptotic bias squared and variance yield the lower asymptotic mean square error. Here we demonstrate the relevance of using the confidence bands for the quantiles using the optima1 number of order statistics on simulated and actua1 data. It appears that if one does not correct for the sign factor the confidence bands are considerably larger. In the financia1 application for the determination of appropriate capita1 buffers usage of the optima1 confidence band implies considerable reduction in capital provisioning. The band without the correction term sometimes requires about 10% more capital vis รก vis the optimal band. Since investment banks nowadays have to provision against such losses by holding capital, .reduction in capital requirements in the order of 10% gives quite a significant reduction in operating costs.
