Estimating Confidence Intervals and Regions for Quantiles in Steady‐State Simulations

ABSTRACT We propose methods based on batching, sectioning, and generalized likelihood ratios (GLRs) for computing confidence intervals (CIs) and confidence regions (CRs) for quantiles in steady‐state simulations. Based on central limit theorems for quantile estimators, CIs and CRs can be computed from a single batch of simulated output using a GLR method to consistently estimate the unknown density function. This paper makes the following contributions: (1) We derive a GLR estimator for distribution sensitivities in the steady‐state setting and, under the geometric‐moment contraction (GMC) conditions, we establish the uniform consistency of the GLR estimators and the asymptotic validity of the respective CIs and CRs for quantiles. (2) We also establish the asymptotic validity of CIs and CRs for quantiles by batching and sectioning methods for steady‐state simulations. Numerical experiments demonstrate the validity of the aforementioned methods as the coverage rates of the CIs and CRs approach the target levels for appropriately large sample sizes. In the steady‐state setting, the sectioning and GLR methods demonstrate their respective advantages in different examples.