Confidence intervals for quantiles when applying variance-reduction techniques

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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.

Original languageEnglish (US)
Article number10
JournalACM Transactions on Modeling and Computer Simulation
Issue number2
StatePublished - Mar 2012

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Computer Science Applications


  • Performance
  • Theory


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