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On the choice of the optimal single order statistic in quantile estimation

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  • Mariusz Bieniek

    (Institute of Mathematics, Maria Curie Skłodowska University)

  • Luiza Pańczyk

    (Institute of Mathematics, Maria Curie Skłodowska University)

Abstract

We study the classical statistical problem of the estimation of quantiles by order statistics of the random sample. For fixed sample size, we determine the single order statistic which is the optimal estimator of a quantile of given order. We propose a totally new approach to the problem, since our optimality criterion is based on the use of nonparametric sharp upper and lower bounds on the bias of the estimation. First, we determine the explicit analytic expressions for the bounds, and then, we choose the order statistic for which the upper and lower bound are simultaneously as close to 0 as possible. The paper contains rigorously proved theoretical results which can be easily implemented in practise. This is also illustrated with numerical examples.

Suggested Citation

  • Mariusz Bieniek & Luiza Pańczyk, 2023. "On the choice of the optimal single order statistic in quantile estimation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(2), pages 303-333, April.
    • Handle: RePEc:spr:aistmt:v:75:y:2023:i:2:d:10.1007_s10463-022-00845-3
    DOI: 10.1007/s10463-022-00845-3
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    References listed on IDEAS

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    1. Nickos Papadatos, 1995. "Maximum variance of order statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(1), pages 185-193, January.
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    3. Mariusz Bieniek, 2007. "Variation diminishing property of densities of uniform generalized order statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 65(3), pages 297-309, May.
    4. R. Kaas & J.M. Buhrman, 1980. "Mean, Median and Mode in Binomial Distributions," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 34(1), pages 13-18, March.
    5. Perrin, Olivier & Redside, Edmond, 2007. "Generalization of Simmons' theorem," Statistics & Probability Letters, Elsevier, vol. 77(6), pages 604-606, March.
    6. Okolewski, Andrzej & Rychlik, Tomasz, 2001. "Sharp distribution-free bounds on the bias in estimating quantiles via order statistics," Statistics & Probability Letters, Elsevier, vol. 52(2), pages 207-213, April.
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