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Smoothed empirical likelihood for the difference of two quantiles with the paired sample

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  • Pangpang Liu

    (Purdue University)

  • Yichuan Zhao

    (Georgia State University)

Abstract

In this paper, we propose a novel smoothed empirical likelihood method for the difference of quantiles with paired samples. While the empirical likelihood for the difference of two quantiles with independent samples has been studied, it is crucial to develop a statistical procedure that accounts for the dependence between paired samples from $${\varvec{X}}=(X_1, X_2)$$ X = ( X 1 , X 2 ) . To this end, we propose two estimating equations for the difference of two quantiles and introduce a nuisance parameter in our smoothed empirical likelihood framework. We demonstrate that our approach yields a limiting distribution that follows the standard $$\chi ^2$$ χ 2 distribution. Extensive simulation studies confirm that our smoothed empirical likelihood method outperforms the normal approximation and method M (Wilcox and Erceg-Hurn in J Appl Stat 39(12):2655–2664, 2012) in most cases. Finally, we illustrate the usefulness of our proposed method by applying it to a real-world data set, estimating the interval of the quantile difference of GDP between different years.

Suggested Citation

  • Pangpang Liu & Yichuan Zhao, 2024. "Smoothed empirical likelihood for the difference of two quantiles with the paired sample," Statistical Papers, Springer, vol. 65(4), pages 2077-2108, June.
  • Handle: RePEc:spr:stpapr:v:65:y:2024:i:4:d:10.1007_s00362-023-01476-3
    DOI: 10.1007/s00362-023-01476-3
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    References listed on IDEAS

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    1. Holger Dette & Jens Wagener & Stanislav Volgushev, 2011. "Comparing Conditional Quantile Curves," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 38(1), pages 63-88, March.
    2. Sergio Firpo, 2007. "Efficient Semiparametric Estimation of Quantile Treatment Effects," Econometrica, Econometric Society, vol. 75(1), pages 259-276, January.
    3. Sun, Yiguo, 2006. "A Consistent Nonparametric Equality Test Of Conditional Quantile Functions," Econometric Theory, Cambridge University Press, vol. 22(4), pages 614-632, August.
    4. Qin, Yong Song, 1997. "Semi-parametric likelihood ratio confidence intervals for various differences of two populations," Statistics & Probability Letters, Elsevier, vol. 33(2), pages 135-143, April.
    5. Miguel A. Delgado & Juan Carlos Escanciano, 2013. "Conditional Stochastic Dominance Testing," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(1), pages 16-28, January.
    6. Yang, Hanfang & Zhao, Yichuan, 2018. "Smoothed jackknife empirical likelihood for the one-sample difference of quantiles," Computational Statistics & Data Analysis, Elsevier, vol. 120(C), pages 58-69.
    7. Hanfang Yang & Yichuan Zhao, 2017. "Smoothed jackknife empirical likelihood for the difference of two quantiles," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(5), pages 1059-1073, October.
    8. Rand R. Wilcox & David M. Erceg-Hurn, 2012. "Comparing two dependent groups via quantiles," Journal of Applied Statistics, Taylor & Francis Journals, vol. 39(12), pages 2655-2664, August.
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