Quantile Estimation When Applying Conditional Monte Carlo

Marvin K. Nakayama

2014

Abstract

We describe how to use conditional Monte Carlo (CMC) to estimate a quantile. CMC is a variance-reduction technique that reduces variance by analytically integrating out some of the variability. We show that the CMC quantile estimator satisfies a central limit theorem and Bahadur representation. We also develop three asymptotically valid confidence intervals (CIs) for a quantile. One CI is based on a finite-difference estimator, another uses batching, and the third applies sectioning. We present numerical results demonstrating the effectiveness of CMC.

References

  1. Asmussen, S. and Glynn, P. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.
  2. Bahadur, R. R. (1966). A note on quantiles in large samples. Annals of Mathematical Statistics, 37:577-580.
  3. Billingsley, P. (1995). Probability and Measure. John Wiley & Sons, New York, third edition.
  4. Bloch, D. A. and Gastwirth, J. L. (1968). On a simple estimate of the reciprocal of the density function. Annals of Mathematical Statistics, 39:1083-1085.
  5. Bofinger, E. (1975). Estimation of a density function using order statistics. Australian Journal of Statistics, 17:1- 7.
  6. Chu, F. and Nakayama, M. K. (2012). Confidence intervals for quantiles when applying variance-reduction techniques. ACM Transactions On Modeling and Computer Simulation, 36:Article 7 (25 pages plus 12-page online-only appendix).
  7. Falk, M. (1986). On the estimation of the quantile density function. Statistics & Probability Letters, 4:69-73.
  8. Fu, M. C., Hong, L. J., and Hu, J.-Q. (2009). Conditional Monte Carlo estimation of quantile sensitivities. Management Science, 55:2019-2027.
  9. Ghosh, J. K. (1971). A new proof of the Bahadur representation of quantiles and an application. Annals of Mathematical Statistics, 42:1957-1961.
  10. Jorion, P. (2007). Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition. McGraw-Hill.
  11. Mood, A. M., Graybill, F. A., and Boes, D. C. (1974). Introduction to the Theory of Statistics. McGraw-Hill, New York, 3rd edition.
  12. Nakayama, M. K. (2014a). Confidence intervals using sectioning for quantiles when applying variancereduction techniques. ACM Transactions on Modeling and Computer Simulation. To appear.
  13. Nakayama, M. K. (2014b). Efficient quantile estimation using conditional Monte Carlo. In preparation.
  14. Ortega, J. M. and Rheinboldt, W. C. (1987). terative Solution of Nonlinear Equations in Several Variables. SIAM.
  15. Ross, S. M. (2006). Simulation. Academic Press, San Diego, CA, fourth edition.
  16. Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York.
  17. U.S. Nuclear Regulatory Commission (1989). Bestestimate calculations of emergency core cooling performance. Nuclear Regulatory Commission Regulatory Guide 1.157, U.S. Nuclear Regulatory Commission, Washington, DC.
  18. U.S. Nuclear Regulatory Commission (2011). Applying statistics. U.S. Nuclear Regulatory Commission Report NUREG-1475, Rev 1, U.S. Nuclear Regulatory Commission, Washington, DC.
Download


Paper Citation


in Harvard Style

K. Nakayama M. (2014). Quantile Estimation When Applying Conditional Monte Carlo . In Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH, ISBN 978-989-758-038-3, pages 280-285. DOI: 10.5220/0005109702800285


in Bibtex Style

@conference{simultech14,
author={Marvin K. Nakayama},
title={Quantile Estimation When Applying Conditional Monte Carlo},
booktitle={Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,},
year={2014},
pages={280-285},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005109702800285},
isbn={978-989-758-038-3},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,
TI - Quantile Estimation When Applying Conditional Monte Carlo
SN - 978-989-758-038-3
AU - K. Nakayama M.
PY - 2014
SP - 280
EP - 285
DO - 10.5220/0005109702800285