Confidence intervals for quantiles when applying variance-reduction techniques

Authors:
Fang Chu;Marvin K. Nakayama
Affiliations:
New Jersey Institute of Technology, Newark, NJ;New Jersey Institute of Technology, Newark, NJ
Venue:
ACM Transactions on Modeling and Computer Simulation (TOMACS)
Year:
2012

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Abstract

Quantiles, which are also known as values-at-risk in finance, frequently arise in practice as measures of risk. This article develops asymptotically valid confidence intervals for quantiles estimated via simulation using variance-reduction techniques (VRTs). We establish our results within a general framework for VRTs, which we show includes importance sampling, stratified sampling, antithetic variates, and control variates. Our method for verifying asymptotic validity is to first demonstrate that a quantile estimator obtained via a VRT within our framework satisfies a Bahadur-Ghosh representation. We then exploit this to show that the quantile estimator obeys a central limit theorem (CLT) and to develop a consistent estimator for the variance constant appearing in the CLT, which enables us to construct a confidence interval. We provide explicit formulae for the estimators for each of the VRTs considered.