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Distribution‐free Approximate Methods for Constructing Confidence Intervals for Quantiles

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  • Chaitra H. Nagaraja
  • Haikady N. Nagaraja

Abstract

Quantile estimation is important for a wide range of applications. While point estimates based on one or two order statistics are common, constructing confidence intervals around them, however, is a more difficult problem. This paper has two goals. First, it surveys the numerous distribution‐free methods for constructing approximate confidence intervals for quantiles. These techniques can be divided roughly into four categories: using a pivotal quantity, resampling, interpolation, and empirical likelihood methods. Second, a method based on the pivotal quantity that has received limited attention in the past is extended. Comprehensive simulation studies are used to compare performance across methods. The proposed method is simple and performs similarly to linear interpolation methods and a smoothed empirical likelihood method. While the proposed method has slightly wider interval widths, it can be calculated for more extreme quantiles even when there are few observations.

Suggested Citation

  • Chaitra H. Nagaraja & Haikady N. Nagaraja, 2020. "Distribution‐free Approximate Methods for Constructing Confidence Intervals for Quantiles," International Statistical Review, International Statistical Institute, vol. 88(1), pages 75-100, April.
    • Handle: RePEc:bla:istatr:v:88:y:2020:i:1:p:75-100
    DOI: 10.1111/insr.12338
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    1. M. Jones, 1992. "Estimating densities, quantiles, quantile densities and density quantiles," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 44(4), pages 721-727, December.
    2. Hettmansperger, Thomas P. & Sheather, Simon J., 1986. "Confidence intervals based on interpolated order statistics," Statistics & Probability Letters, Elsevier, vol. 4(2), pages 75-79, March.
    3. Wang Zhou & Bing-Yi Jing, 2003. "Adjusted empirical likelihood method for quantiles," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(4), pages 689-703, December.
    4. Goldman, Matt & Kaplan, David M., 2017. "Fractional order statistic approximation for nonparametric conditional quantile inference," Journal of Econometrics, Elsevier, vol. 196(2), pages 331-346.
    5. Larocque, Denis & Randles, Ronald H., 2008. "Confidence Intervals for a Discrete Population Median," The American Statistician, American Statistical Association, vol. 62, pages 32-39, February.
    6. Kaplan, David M., 2015. "Improved quantile inference via fixed-smoothing asymptotics and Edgeworth expansion," Journal of Econometrics, Elsevier, vol. 185(1), pages 20-32.
    7. Koehler, Elizabeth & Brown, Elizabeth & Haneuse, Sebastien J.-P. A., 2009. "On the Assessment of Monte Carlo Error in Simulation-Based Statistical Analyses," The American Statistician, American Statistical Association, vol. 63(2), pages 155-162.
    8. Jesse Frey & Yimin Zhang, 2017. "What Do Interpolated Nonparametric Confidence Intervals for Population Quantiles Guarantee?," The American Statistician, Taylor & Francis Journals, vol. 71(4), pages 305-309, October.
    9. Nyblom, Jukka, 1992. "Note on interpolated order statistics," Statistics & Probability Letters, Elsevier, vol. 14(2), pages 129-131, May.
    10. Alan Hutson, 1999. "Calculating nonparametric confidence intervals for quantiles using fractional order statistics," Journal of Applied Statistics, Taylor & Francis Journals, vol. 26(3), pages 343-353.
    11. Olivier J. M. Guilbaud, 2018. "Some Complementary History and Results," The American Statistician, Taylor & Francis Journals, vol. 72(3), pages 300-301, July.
    12. H. Nagaraja & Karthik Bharath & Fangyuan Zhang, 2015. "Spacings around an order statistic," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(3), pages 515-540, June.
    13. Sheather, Simon J. & McKean, Joseph W., 1987. "A comparison of testing and confidence interval methods for the median," Statistics & Probability Letters, Elsevier, vol. 6(1), pages 31-36, September.
    14. Athanassios N. Avramidis & James R. Wilson, 1998. "Correlation-Induction Techniques for Estimating Quantiles in Simulation Experiments," Operations Research, INFORMS, vol. 46(4), pages 574-591, August.
    15. Michael Falk, 1989. "A note on uniform asymptotic normality of intermediate order statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 41(1), pages 19-29, March.
    16. Paul Glasserman & Philip Heidelberger & Perwez Shahabuddin, 2000. "Variance Reduction Techniques for Estimating Value-at-Risk," Management Science, INFORMS, vol. 46(10), pages 1349-1364, October.
    17. Xing Jin & Michael C. Fu & Xiaoping Xiong, 2003. "Probabilistic Error Bounds for Simulation Quantile Estimators," Management Science, INFORMS, vol. 49(2), pages 230-246, February.
    18. Bruce Schmeiser, 1982. "Batch Size Effects in the Analysis of Simulation Output," Operations Research, INFORMS, vol. 30(3), pages 556-568, June.
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    1. Chaitra H. Nagaraja & Haikady N. Nagaraja, 2020. "Correction to ‘Distribution‐free Approximate Methods for Constructing Confidence Intervals for Quantiles’," International Statistical Review, International Statistical Institute, vol. 88(2), pages 519-519, August.

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