Quantiles, as a performance measure, arise in many practical contexts. In finance, quantiles are called values-at-risk (VARs), and they are widely used in the financial industry to measure portfolio risk. When the cumulative distribution function is unknown, the quantile can not be computed exactly and must be estimated. In addition to computing a point estimate for the quantile, it is important to also provide a confidence interval for the quantile as a way of indicating the error in the estimate. A problem with crude Monte Carlo is that the resulting confidence interval may be large, which is often the case when estimating extreme quantiles. This motivates applying variance-reduction techniques (VRTs) to try to obtain more efficient quantile estimators. Much of the previous work on estimating quantiles using VRTs did not provide methods for constructing asymptotically valid confidence intervals.
This research developed asymptotically valid confidence intervals for quantiles that are estimated using simulation with VRTs. The VRTs considered were importance sampling (IS), stratified sampling (SS), antithetic variates (AV), and control variates (CV). The method of proving the asymptotic validity was to first show that the quantile estimators obtained with VRTs satisfies a Bahadur-Ghosh representation. Then this was employed to prove central limit theorems (CLTs) and to obtain consistent estimators of the variances in the CLTs, which were used to construct confidence intervals. After the theoretical framework was established, explicit algorithms were presented to construct confidence intervals for quantiles when applying IS+SS, AV and CV. An empirical study of the finite-sample behavior of the confidence intervals was also performed on two stochastic models: a standard normal/bivariate normal distribution and a stochastic activity network (SAN)
