{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T23:29:52Z","timestamp":1768692592158,"version":"3.49.0"},"reference-count":36,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2012,3,1]],"date-time":"2012-03-01T00:00:00Z","timestamp":1330560000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/100000147","name":"Division of Civil, Mechanical and Manufacturing Innovation","doi-asserted-by":"publisher","award":["CMMI-0926949"],"award-info":[{"award-number":["CMMI-0926949"]}],"id":[{"id":"10.13039\/100000147","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Model. Comput. Simul."],"published-print":{"date-parts":[[2012,3]]},"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.1145\/2133390.2133394","type":"journal-article","created":{"date-parts":[[2012,3,27]],"date-time":"2012-03-27T15:17:31Z","timestamp":1332861451000},"page":"1-25","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":41,"title":["Confidence intervals for quantiles when applying variance-reduction techniques"],"prefix":"10.1145","volume":"22","author":[{"given":"Fang","family":"Chu","sequence":"first","affiliation":[{"name":"New Jersey Institute of Technology, Newark, NJ"}]},{"given":"Marvin K.","family":"Nakayama","sequence":"additional","affiliation":[{"name":"New Jersey Institute of Technology, Newark, NJ"}]}],"member":"320","published-online":{"date-parts":[[2012,3,30]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1287\/opre.46.4.574"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-7152(86)90076-3"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177699450"},{"key":"e_1_2_2_4_1","volume-title":"Probability and Measure","author":"Billingsley P.","unstructured":"Billingsley , P. 1995. Probability and Measure , 3 rd Ed. John Wiley & Sons , New York . Billingsley, P. 1995. Probability and Measure, 3rd Ed. John Wiley & Sons, New York.","edition":"3"},{"key":"e_1_2_2_5_1","volume-title":"Convergence of Probability Measures","author":"Billingsley P.","unstructured":"Billingsley , P. 1999. Convergence of Probability Measures , 2 nd Ed. John Wiley & Sons , New York . Billingsley, P. 1999. Convergence of Probability Measures, 2nd Ed. John Wiley & Sons, New York.","edition":"2"},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177698342"},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-842X.1975.tb01366.x"},{"key":"e_1_2_2_9_1","volume-title":"Proceedings of the Winter Simulation Conference. IEEE, 2751--2761","author":"Chu F.","unstructured":"Chu , F. and Nakayama , M. K . 2010. Confidence intervals for quantiles and value-at-risk when applying importance sampling . In Proceedings of the Winter Simulation Conference. IEEE, 2751--2761 . Chu, F. and Nakayama, M. K. 2010. Confidence intervals for quantiles and value-at-risk when applying importance sampling. In Proceedings of the Winter Simulation Conference. IEEE, 2751--2761."},{"key":"e_1_2_2_10_1","doi-asserted-by":"crossref","unstructured":"David H. A. and Nagaraja H. 2003. Order Statistics 3rd Ed. Wiley Hoboken NJ. David H. A. and Nagaraja H. 2003. Order Statistics 3rd Ed. Wiley Hoboken NJ.","DOI":"10.1002\/0471722162"},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.3905\/jod.1997.407971"},{"key":"e_1_2_2_12_1","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.1090.1090"},{"key":"e_1_2_2_13_1","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177693063"},{"key":"e_1_2_2_14_1","volume-title":"Monte Carlo Methods in Financial Engineering","author":"Glasserman P.","unstructured":"Glasserman , P. 2004. Monte Carlo Methods in Financial Engineering . Springer , New York . Glasserman, P. 2004. Monte Carlo Methods in Financial Engineering. Springer, New York."},{"key":"e_1_2_2_15_1","volume-title":"Proceedings of the 6th International Conference on Computational Finance. Y. S. Abu-Mostafa, B. L. Baron, A. W. Lo, and A. S. Weigend, Eds., 7--24","author":"Glasserman P.","unstructured":"Glasserman , P. , Heidelberger , P. , and Shahabuddin , P . 2000a. Importance sampling and stratification for value-at-risk . In Proceedings of the 6th International Conference on Computational Finance. Y. S. Abu-Mostafa, B. L. Baron, A. W. Lo, and A. S. Weigend, Eds., 7--24 . Glasserman, P., Heidelberger, P., and Shahabuddin, P. 2000a. Importance sampling and stratification for value-at-risk. In Proceedings of the 6th International Conference on Computational Finance. Y. S. Abu-Mostafa, B. L. Baron, A. W. Lo, and A. S. Weigend, Eds., 7--24."},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.46.10.1349.12274"},{"key":"e_1_2_2_17_1","volume-title":"Proceedings of the 2nd St. Petersburg Workshop on Simulation. Publishing House of St","author":"Glynn P. W.","year":"1996","unstructured":"Glynn , P. W. 1996 . Importance sampling for Monte Carlo estimation of quantiles . In Proceedings of the 2nd St. Petersburg Workshop on Simulation. Publishing House of St . Petersburg University, St. Petersburg, Russia, 180--185. Glynn, P. W. 1996. Importance sampling for Monte Carlo estimation of quantiles. In Proceedings of the 2nd St. Petersburg Workshop on Simulation. Publishing House of St. Petersburg University, St. Petersburg, Russia, 180--185."},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.35.11.1367"},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/1015467.1015478"},{"key":"e_1_2_2_20_1","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1111\/j.2517-6161.1988.tb01735.x","article-title":"On the distribution of a Studentized quantile","volume":"50","author":"Hall P.","year":"1988","unstructured":"Hall , P. and Sheather , S. J. 1988 . On the distribution of a Studentized quantile . J. Roy. Statist. Soc. B 50 , 381 -- 391 . Hall, P. and Sheather, S. J. 1988. On the distribution of a Studentized quantile. J. Roy. Statist. Soc. B 50, 381--391.","journal-title":"J. Roy. Statist. Soc. B"},{"key":"e_1_2_2_21_1","volume-title":"A Course in Pure Mathematics","author":"Hardy G. H.","unstructured":"Hardy , G. H. 1952. A Course in Pure Mathematics , 10 th Ed. Cambridge University Press , Cambridge, UK . Hardy, G. H. 1952. A Course in Pure Mathematics, 10th Ed. Cambridge University Press, Cambridge, UK.","edition":"10"},{"key":"e_1_2_2_22_1","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.44.9.1295"},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1287\/opre.1080.0531"},{"key":"e_1_2_2_24_1","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.36.7.835"},{"key":"e_1_2_2_25_1","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.49.2.230.12743"},{"key":"e_1_2_2_26_1","doi-asserted-by":"publisher","DOI":"10.1145\/1225275.1225278"},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177698690"},{"key":"e_1_2_2_28_1","volume-title":"Elements of Large-Sample Theory","author":"Lehmann E. L.","unstructured":"Lehmann , E. L. 1999. Elements of Large-Sample Theory . Springer , New York . Lehmann, E. L. 1999. Elements of Large-Sample Theory. Springer, New York."},{"key":"e_1_2_2_29_1","doi-asserted-by":"publisher","DOI":"10.1002\/nav.20358"},{"key":"e_1_2_2_30_1","volume-title":"Smoothing Techniques for Curve Estimation","author":"Parzen E.","unstructured":"Parzen , E. 1979. Density quantile estimation approach to statistical data modelling . In Smoothing Techniques for Curve Estimation , Springer , Berlin . Parzen, E. 1979. Density quantile estimation approach to statistical data modelling. In Smoothing Techniques for Curve Estimation, Springer, Berlin."},{"key":"e_1_2_2_31_1","volume-title":"Limit Theorems in Probability Theory","author":"Petrov V. V.","unstructured":"Petrov , V. V. 1995. Limit Theorems in Probability Theory . Oxford University Press , Oxford, UK . Petrov, V. V. 1995. Limit Theorems in Probability Theory. Oxford University Press, Oxford, UK."},{"key":"e_1_2_2_32_1","volume-title":"Simulation","author":"Ross S.","unstructured":"Ross , S. 1997. Simulation , 2 nd Ed. Academic Press , San Diego, CA . Ross, S. 1997. Simulation, 2nd Ed. Academic Press, San Diego, CA.","edition":"2"},{"key":"e_1_2_2_33_1","volume-title":"Principles of Mathematical Analysis","author":"Rudin W.","unstructured":"Rudin , W. 1976. Principles of Mathematical Analysis , 3 rd Ed. McGraw-Hill , New York . Rudin, W. 1976. Principles of Mathematical Analysis, 3rd Ed. McGraw-Hill, New York.","edition":"3"},{"key":"e_1_2_2_34_1","volume-title":"Approximation Theorems of Mathematical Statistics","author":"Serfling R. J.","unstructured":"Serfling , R. J. 1980. Approximation Theorems of Mathematical Statistics . John Wiley & Sons , New York . Serfling, R. J. 1980. Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York."},{"key":"e_1_2_2_35_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.orl.2010.02.007"},{"key":"e_1_2_2_36_1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.53.1.127"},{"key":"e_1_2_2_37_1","doi-asserted-by":"crossref","unstructured":"Wand M. P. and Jones M. C. 1995. Kernel Smoothing. Chapman & Hall London. Wand M. P. and Jones M. C. 1995. Kernel Smoothing. Chapman & Hall London.","DOI":"10.1007\/978-1-4899-4493-1"}],"container-title":["ACM Transactions on Modeling and Computer Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2133390.2133394","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2133390.2133394","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T10:06:05Z","timestamp":1750241165000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2133390.2133394"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,3]]},"references-count":36,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2012,3]]}},"alternative-id":["10.1145\/2133390.2133394"],"URL":"https:\/\/doi.org\/10.1145\/2133390.2133394","relation":{},"ISSN":["1049-3301","1558-1195"],"issn-type":[{"value":"1049-3301","type":"print"},{"value":"1558-1195","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,3]]},"assertion":[{"value":"2010-07-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-09-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-03-30","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}