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Calculating nonparametric confidence intervals for quantiles using fractional order statistics

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  • Alan Hutson

Abstract

In this paper, we provide an easy-to-program algorithm for constructing the preselected 100(1 - alpha)% nonparametric confidence interval for an arbitrary quantile, such as the median or quartile, by approximating the distribution of the linear interpolation estimator of the quantile function Q L ( u ) = (1 - epsilon) X \[ n u ] + epsilon X \[ n u ] + 1 with the distribution of the fractional order statistic Q I ( u ) = Xn u , as defined by Stigler, where n = n + 1 and \[ . ] denotes the floor function. A simulation study verifies the accuracy of the coverage probabilities. An application to the extreme-value problem in flood data analysis in hydrology is illustrated.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:japsta:v:26:y:1999:i:3:p:343-353
    DOI: 10.1080/02664769922458
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    References listed on IDEAS

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    1. Mudholkar, Govind S. & Hutson, Alan D., 1997. "Some refinements of the quasi-quantiles," Statistics & Probability Letters, Elsevier, vol. 32(1), pages 67-74, February.
    2. Nyblom, Jukka, 1992. "Note on interpolated order statistics," Statistics & Probability Letters, Elsevier, vol. 14(2), pages 129-131, May.
    3. 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.
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    Cited by:

    1. Farkas, Walter & Fringuellotti, Fulvia & Tunaru, Radu, 2020. "A cost-benefit analysis of capital requirements adjusted for model risk," Journal of Corporate Finance, Elsevier, vol. 65(C).
    2. David M. Kaplan, 2013. "IDEAL Inference on Conditional Quantiles via Interpolated Duals of Exact Analytic L-statistics," Working Papers 1316, Department of Economics, University of Missouri.
    3. David M. Kaplan & Lonnie Hofmann, 2019. "High-order coverage of smoothed Bayesian bootstrap intervals for population quantiles," Working Papers 1914, Department of Economics, University of Missouri, revised 19 Sep 2020.
    4. Ralda Nehme & Olli Pietiläinen & Mykyta Artomov & Matthew Tegtmeyer & Vera Valakh & Leevi Lehtonen & Christina Bell & Tarjinder Singh & Aditi Trehan & John Sherwood & Danielle Manning & Emily Peirent , 2022. "The 22q11.2 region regulates presynaptic gene-products linked to schizophrenia," Nature Communications, Nature, vol. 13(1), pages 1-21, December.
    5. David M. Kaplan & Matt Goldman, 2013. "IDEAL Quantile Inference via Interpolated Duals of Exact Analytic L-statistics," Working Papers 1315, Department of Economics, University of Missouri.
    6. Matt Goldman & David M. Kaplan, 2018. "Non‐parametric inference on (conditional) quantile differences and interquantile ranges, using L‐statistics," Econometrics Journal, Royal Economic Society, vol. 21(2), pages 136-169, June.
    7. Alan Hutson, 2000. "A composite quantile function estimator with applications in bootstrapping," Journal of Applied Statistics, Taylor & Francis Journals, vol. 27(5), pages 567-577.
    8. Kaplan, David M., 2015. "Improved quantile inference via fixed-smoothing asymptotics and Edgeworth expansion," Journal of Econometrics, Elsevier, vol. 185(1), pages 20-32.
    9. Maria E. Frey & Hans C. Petersen & Oke Gerke, 2020. "Nonparametric Limits of Agreement for Small to Moderate Sample Sizes: A Simulation Study," Stats, MDPI, vol. 3(3), pages 1-13, August.
    10. 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.
    11. Alan Hutson, 2018. "Comment on “What Do Interpolated Nonparametric Confidence Intervals for Population Quantiles Guarantee?”, Frey and Zhang (2017)," The American Statistician, Taylor & Francis Journals, vol. 72(3), pages 302-302, July.
    12. Wang, Wei & Cammi, Antonio & Di Maio, Francesco & Lorenzi, Stefano & Zio, Enrico, 2018. "A Monte Carlo-based exploration framework for identifying components vulnerable to cyber threats in nuclear power plants," Reliability Engineering and System Safety, Elsevier, vol. 175(C), pages 24-37.
    13. Reyes-Fuentes, Melisa & del-Valle-Gallegos, Edmundo & Duran-Gonzalez, Julian & Ortíz-Villafuerte, Javier & Castillo-Durán, Rogelio & Gómez-Torres, Armando & Queral, Cesar, 2021. "AZTUSIA: A new application software for Uncertainty and Sensitivity analysis for nuclear reactors," Reliability Engineering and System Safety, Elsevier, vol. 209(C).
    14. Szilárd Nemes, 2019. "Likelihood Confidence Intervals When Only Ranges Are Available," Stats, MDPI, vol. 2(1), pages 1-7, February.
    15. Hutson, Alan D., 2000. "Estimating the covariance of bivariate order statistics with applications," Statistics & Probability Letters, Elsevier, vol. 48(2), pages 195-203, June.
    16. Hund, Lauren & Schroeder, Benjamin & Rumsey, Kellin & Huerta, Gabriel, 2018. "Distinguishing between model- and data-driven inferences for high reliability statistical predictions," Reliability Engineering and System Safety, Elsevier, vol. 180(C), pages 201-210.
    17. Jones, M. C., 2002. "On fractional uniform order statistics," Statistics & Probability Letters, Elsevier, vol. 58(1), pages 93-96, May.
    18. 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.
    19. Sanchez-Saez, F. & Sánchez, A.I. & Villanueva, J.F. & Carlos, S. & Martorell, S., 2018. "Uncertainty analysis of a large break loss of coolant accident in a pressurized water reactor using non-parametric methods," Reliability Engineering and System Safety, Elsevier, vol. 174(C), pages 19-28.
    20. Su, Steve, 2009. "Confidence intervals for quantiles using generalized lambda distributions," Computational Statistics & Data Analysis, Elsevier, vol. 53(9), pages 3324-3333, July.

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