Empirical Regression Quantile Processes


We address the problem of estimating quantile-based statistical functionals, when the measured or controlled entities depend on exogenous variables which are not under our control. As a suitable tool we propose the empirical process of the average regression quantiles. It partially masks the effect of covariates and has other properties convenient for applications, e.g. for coherent risk measures of various types in the situations with covariates.

This is a preview of subscription content, log in to check access.


  1. [1]

    G. W. Bassett, Jr.: A property of the observations fit by the extreme regression quantiles. Comput. Stat. Data Anal. 6 (1988), 353–359.

    MathSciNet  Article  Google Scholar 

  2. [2]

    G. W. Bassett, Jr., R. Koenker: An empirical quantile function for linear models with iid errors. J. Am. Stat. Assoc. 77 (1982), 405–415.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    C. Gutenbrunner, J. Jurečková: Regression rank scores and regression quantiles. Ann. Stat. 20 (1992), 305–330.

    MathSciNet  Article  Google Scholar 

  4. [4]

    J. Hájek: Extension of the Kolmogorov-Smirnov test to regression alternatives. Proceedings of the International Research Seminar (L. LeCam, ed.). University of California Press, Berkeley, 1965, pp. 45–60.

    Google Scholar 

  5. [5]

    L. A. Jaeckel: Estimating regression coefficients by minimizing the dispersion of the residuals. Ann. Math. Stat. 43 (1972), 1449–1458.

    MathSciNet  Article  Google Scholar 

  6. [6]

    J. Jurečková: Averaged extreme regression quantile. Extremes 19 (2016), 41–49.

    MathSciNet  Article  Google Scholar 

  7. [7]

    J. Jurečková, J. Picek: Two-step regression quantiles. Sankhyā 67 (2005), 227–252.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    J. Jurečková, J. Picek: Averaged regression quantiles. Contemporary Developments in Statistical Theory (S. Lahiri et al., eds.). Springer Proceedings in Mathematics and Statistics 68, Springer, Cham, 2014, pp. 203–216.

    Google Scholar 

  9. [9]

    J. Jurečková, J. Picek, M. Schindler: Robust Statistical Methods With R. CRC Press, Boca Raton, 2019.

    Google Scholar 

  10. [10]

    J. Jurečková, P. K. Sen, J. Picek: Methodology in Robust and Nonparametric Statistics. CRC Press, Boca Raton, 2013.

    Google Scholar 

  11. [11]

    J. D. Kloke, J. W. McKean: Rfit: Rank-based estimation for linear models. The R Journal 4 (2012), 57–64.

    Article  Google Scholar 

  12. [12]

    R. Koenker: Quantile Regression. Econometric Society Monographs 38, Cambridge University Press, Cambridge, 2005.

    Google Scholar 

  13. [13]

    R. Koenker: quantreg: Quantile Regression. R package version 5.51. Available at https://CRAN.R-project.org/package=quantreg.

  14. [14]

    R. Koenker, G. Bassett, Jr.: Regression quantiles. Econometrica 46 (1978), 33–50.

    MathSciNet  Article  Google Scholar 

  15. [15]

    S. Portnoy: Tightness of the sequence of empiric c.d.f. processes defined from regression fractiles. Robust and Nonlinear Time Series Analysis (J. Franke et al., eds.). Lecture Notes in Statistics 26, Springer, New York, 1984, pp. 231–245.

    Google Scholar 

  16. [16]

    S. Portnoy: Asymptotic behavior of the number of regression quantile breakpoints. SIAM J. Sci. Stat. Comput. 12 (1991), 867–883.

    MathSciNet  Article  Google Scholar 

  17. [17]

    D. Ruppert, R. J. Carroll: Trimmed least squares estimation in the linear model. J. Am. Stat. Assoc. 75 (1980), 828–838.

    MathSciNet  Article  Google Scholar 

Download references


We thank the reviewer for careful reading of our manuscript and for valuable comments.

Author information



Corresponding author

Correspondence to Martin Schindler.

Additional information

The authors gratefully acknowledge the support of the Grant GACR 18-01137S.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jurečková, J., Picek, J. & Schindler, M. Empirical Regression Quantile Processes. Appl Math 65, 257–269 (2020). https://doi.org/10.21136/AM.2020.0295-19

Download citation


  • averaged regression quantile
  • one-step regression quantile
  • R-estimator
  • functionals of the quantile process

MSC 2020

  • 62J02
  • 62G30
  • 90C05
  • 65K05
  • 49M29