Estimating Confidence Intervals and Regions for Quantiles by Monte Carlo Simulation

Abstract

We propose methods for computing confidence intervals (CIs) and confidence regions (CRs) for quantiles in finite-horizon and steady-state simulations. Based on central limit theorems for quantile estimators, CIs and CRs for quantiles can be computed from a single-batch of simulated output using a generalized likelihood ratio (GLR) method to consistently estimate the unknown density function. We derive a GLR estimator for distribution sensitivities in the steady-state setting and establish a uniform consistency of the GLR estimators under a geometric-moment contraction (GMC) condition. We establish the asymptotic validity of CIs and CRs for quantiles by the GLR method in steady-state simulations, and the asymptotic validity of CIs and CRs for quantiles by batching and sectioning methods in finite-horizon simulation. Numerical experiments demonstrate the advantage of the GLR method over the batching and sectioning methods in estimating CRs for quantiles when the dimension of the vector of quantiles is large and the sample size is relatively small.

Keywords: Monte Carlo simulation, quantiles, confidence intervals, confidence regions, sensitivity analysis