Estimation of extreme quantiles for functions of dependent random variables

We propose a new method for estimating the extreme quantiles for a function of several dependent random variables. In contrast with the conventional approach based on extreme value theory, we do not impose the condition that the tail of the underlying distribution admits an approximate parametric form, and, furthermore, our estimation makes use of the full observed data. The method proposed is semiparametric as no parametric forms are assumed on the marginal distributions. But we select appropriate bivariate copulas to model the joint dependence structure by taking advantage of the recent development in constructing large dimensional vine copulas. Consequently a sample quantile resulting from a large bootstrap sample drawn from the fitted joint distribution is taken as the estimator for the extreme quantile. This estimator is proved to be consistent under the regularity conditions on the closeness between a quantile set and its truncated set, and the empirical approximation for the truncated set. The simulation results lend further support to the reliable and robust performance of the method proposed. The method is further illustrated by a real world example in backtesting financial risk models.National Social Science Foundation of China [12CTJ007]; Simons Foundation; Engineering and Physical Sciences Research CouncilSCI(E)ARTICLEq.yao@lse.ac.uk51001-10247