Quantile Estimation with Latin Hypercube Sampling

Quantiles are often used to measure risk of stochastic systems. We examine quantile estimators obtained using simulation with Latin hypercube sampling (LHS), a variance-reduction technique that efficiently extends stratified sampling to higher dimensions and produces negatively correlated outputs. We consider single-sample LHS (ssLHS), which minimizes the variance that can be obtained from LHS, and also replicated LHS (rLHS). We develop a consistent estimator of the asymptotic variance of the ssLHS quantile estimator’s central limit theorem, enabling us to provide the first confidence interval (CI) for a quantile when applying ssLHS. For rLHS, we construct CIs using batching and sectioning. On average, our rLHS CIs are shorter than previous rLHS CIs and only slightly wider than the ssLHS CI. We establish the asymptotic validity of the CIs by first proving that the quantile estimators satisfy Bahadur representations, which show that the quantile estimators can be approximated by linear transformations of estimators of the cumulative distribution function. We present numerical results comparing the various CIs.