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Quantile Estimation in Dependent Sequences

Author

Listed:
  • P. Heidelberger

    (IBM Thomas J. Watson Research Center, Yorktown Heights, New York)

  • P. A. W. Lewis

    (Naval Postgraduate School, Monterey, California)

Abstract

Standard nonparametric estimators of quantiles based on order statistics can be used not only when the data are i.i.d., but also when the data are drawn from a stationary, ϕ-mixing process of continuous random variables. However, when the random variables are highly positively correlated, the sample sizes needed for estimating extreme quantiles become computationally unmanageable. This paper gives a practical scheme, based on a maximum transformation in a two-way layout of the data, that reduces the sample size sufficiently to allow an experimenter to obtain a point estimate of an extreme quantile. The paper gives three schemes that lead to confidence interval estimates for the quantile. One uses a spectral analysis of the reduced sample. The other two, averaged group quantiles and nested group quantiles, are extensions of the method of batched means to quantile estimation. These two schemes give even greater data compaction than the first scheme. None of the schemes requires that the process being simulated is regenerative.

Suggested Citation

  • P. Heidelberger & P. A. W. Lewis, 1984. "Quantile Estimation in Dependent Sequences," Operations Research, INFORMS, vol. 32(1), pages 185-209, February.
    • Handle: RePEc:inm:oropre:v:32:y:1984:i:1:p:185-209
    DOI: 10.1287/opre.32.1.185
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    Citations

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    Cited by:

    1. Guangwu Liu & Liu Jeff Hong, 2009. "Kernel estimation of quantile sensitivities," Naval Research Logistics (NRL), John Wiley & Sons, vol. 56(6), pages 511-525, September.
    2. L. Jeff Hong, 2009. "Estimating Quantile Sensitivities," Operations Research, INFORMS, vol. 57(1), pages 118-130, February.
    3. Shane G. Henderson & Peter W. Glynn, 2001. "Computing Densities for Markov Chains via Simulation," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 375-400, May.
    4. Batur, D. & Choobineh, F., 2010. "A quantile-based approach to system selection," European Journal of Operational Research, Elsevier, vol. 202(3), pages 764-772, May.
    5. Chen, E. Jack & Kelton, W. David, 2006. "Quantile and tolerance-interval estimation in simulation," European Journal of Operational Research, Elsevier, vol. 168(2), pages 520-540, January.
    6. Christos Alexopoulos & David Goldsman & Anup C. Mokashi & Kai-Wen Tien & James R. Wilson, 2019. "Sequest: A Sequential Procedure for Estimating Quantiles in Steady-State Simulations," Operations Research, INFORMS, vol. 67(4), pages 1162-1183, July.
    7. Park, Dae S. & Kim, Yun B. & Shin, Key I. & Willemain, Thomas R., 2001. "Simulation output analysis using the threshold bootstrap," European Journal of Operational Research, Elsevier, vol. 134(1), pages 17-28, October.
    8. Demet Batur & F. Fred Choobineh, 2021. "Selecting the Best Alternative Based on Its Quantile," INFORMS Journal on Computing, INFORMS, vol. 33(2), pages 657-671, May.
    9. Batur, D. & Choobineh, F., 2012. "Stochastic dominance based comparison for system selection," European Journal of Operational Research, Elsevier, vol. 220(3), pages 661-672.
    10. Mingbin Ben Feng & Eunhye Song, 2020. "Optimal Nested Simulation Experiment Design via Likelihood Ratio Method," Papers 2008.13087, arXiv.org, revised Jul 2021.
    11. L. Jeff Hong & Guangwu Liu, 2010. "Pathwise Estimation of Probability Sensitivities Through Terminating or Steady-State Simulations," Operations Research, INFORMS, vol. 58(2), pages 357-370, April.
    12. Shuzhen Yang, 2021. "Compensatory model for quantile estimation and application to VaR," Papers 2112.07278, arXiv.org.
    13. Wei Jiang & Steven Kou, 2021. "Simulating risk measures via asymptotic expansions for relative errors," Mathematical Finance, Wiley Blackwell, vol. 31(3), pages 907-942, July.

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