Computational Statistics

, Volume 31, Issue 3, pp 829–844 | Cite as

The shooting S-estimator for robust regression

  • Viktoria Öllerer
  • Andreas Alfons
  • Christophe Croux
Original Paper


To perform multiple regression, the least squares estimator is commonly used. However, this estimator is not robust to outliers. Therefore, robust methods such as S-estimation have been proposed. These estimators flag any observation with a large residual as an outlier and downweight it in the further procedure. However, a large residual may be caused by an outlier in only one single predictor variable, and downweighting the complete observation results in a loss of information. Therefore, we propose the shooting S-estimator, a regression estimator that is especially designed for situations where a large number of observations suffer from contamination in a small number of predictor variables. The shooting S-estimator combines the ideas of the coordinate descent algorithm with simple S-regression, which makes it robust against componentwise contamination, at the cost of failing the regression equivariance property.


Cellwise outliers Componentwise contamination Shooting algorithm Coordinate descent algorithm Regression S-estimation 



We gratefully acknowledge support from the GOA/12/014 Project of the Research Fund KU Leuven. We thank the referees for their constructive comments, and in particular the third anonymous referee who corrected some flaws in the first version of the paper and who made many suggestions for improving the write up of the paper.


  1. Alfons A, Croux C, Gelper S (2013) Sparse least trimmed squares regression for analyzing high-dimensional large data sets. Ann Appl Stat 7(1):226–248MathSciNetCrossRefzbMATHGoogle Scholar
  2. Alqallaf F, Van Aelst S, Yohai V, Zamar R (2009) Propagation of outliers in multivariate data. Ann Stat 37(1):311–331MathSciNetCrossRefzbMATHGoogle Scholar
  3. Belsley D, Kuh E, Welsch R (1980) Regression diagnostics: identifying influential data and source of collinearity. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  4. Brown P (1982) Multivariate calibration. J R Stat Soc Ser B 44(3):287–321MathSciNetzbMATHGoogle Scholar
  5. Friedman J, Hastie T, Hofling H, Tibshirani R (2007) Pathwise coordinate optimization. Ann Appl Stat 1(2):302–332MathSciNetCrossRefzbMATHGoogle Scholar
  6. Fu W (1998) Penalized regressions: the bridge versus the lasso. J Comput Graph Stat 7(3):397–416MathSciNetGoogle Scholar
  7. Harrison D, Rubinfeld D (1978) Hedonic housing prices and the demand of clean air. J Environ Econ Manag 5(1):81–102CrossRefzbMATHGoogle Scholar
  8. Koller M, Stahel W (2011) Sharpening Wald-type inference in robust regression for small samples. Comput Stat Data Anal 55(8):2504–2515MathSciNetCrossRefGoogle Scholar
  9. Little R (1992) Regression with missing X’s: a review. J Am Stat Assoc 87(420):1227–1237Google Scholar
  10. Maronna R, Martin R, Yohai V (2006) Robust statistics, 2nd edn. Wiley, HobokenCrossRefzbMATHGoogle Scholar
  11. Rousseeuw P, Leroy A (1987) Robust regression and outlier detection. Wiley, HobokenCrossRefzbMATHGoogle Scholar
  12. StataCorp (2013) Stata: release 13. Stata Press, College Station, Texas, Statistical SoftwareGoogle Scholar
  13. Tseng P (2001) Convergence of a block coordinate descent method for nondifferentiable minimization. J Optim Theory Appl 3:475–494MathSciNetCrossRefzbMATHGoogle Scholar
  14. Van Aelst S, Vandervieren E, Willems G (2010) Robust principal component analysis based on pairwise correlation estimators. In: Lechevallier Y, Saporta G (eds) COMPSTAT 2010: proceedings in computational statistics. Physika, Heidelberg, pp 1677–1684Google Scholar
  15. Van Aelst S, Vandervieren E, Willems G (2011) Stahel–Donoho estimators with cellwise weights. J Stat Comput Simul 81(1):1–27MathSciNetCrossRefzbMATHGoogle Scholar
  16. Van Aelst S, Vandervieren E, Willems G (2012) A Stahel–Donoho estimator based on huberized outlyingness. Comput Stat Data Anal 56(3):531–542MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Viktoria Öllerer
    • 1
  • Andreas Alfons
    • 2
  • Christophe Croux
    • 1
  1. 1.Faculty of Economics and BusinessKU LeuvenLouvainBelgium
  2. 2.Erasmus School of EconomicsErasmus University RotterdamRotterdamThe Netherlands

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