Robustness of the deepest projection regression functional

  • Yijun ZuoEmail author
Regular Article


Depth notions in regression have been systematically proposed and examined in Zuo (arXiv:1805.02046, 2018). One of the prominent advantages of the notion of depth is that it can be directly utilized to introduce median-type deepest estimating functionals (or estimators in the case of empirical distributions) for location or regression parameters in a multi-dimensional setting. Regression depth shares the advantage. Depth induced deepest estimating functionals are expected to inherit desirable and inherent robustness properties (e.g. bounded maximum bias and influence function and high breakdown point) as their univariate location counterpart does. Investigating and verifying the robustness of the deepest projection estimating functional (in terms of maximum bias, asymptotic and finite sample breakdown point, and influence function) is the major goal of this article. It turns out that the deepest projection estimating functional possesses a bounded influence function and the best possible asymptotic breakdown point as well as the best finite sample breakdown point with robust choice of its univariate regression and scale component.


Depth Linear regression Deepest regression estimating functionals Maximum bias Breakdown point Influence function Robustness 

Mathematics Subject Classification

Primary 62G05 Secondary 62G08 62G35 62G30 



The author thanks Professor Emeritus James Stapleton for his careful English proofreading and an anonymous referee who provided insightful comments and suggestions which have led to significant improvements of the manuscript.


  1. Adrover J, Yohai VJ (2002) Projection estimates of multivariate location. Ann Stat 30:1760–1781MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bai ZD, He X (1999) Asymptotic distributions of the maximal depth regression and multivariate location. Ann Stat 27(5):1616–1637 577-580MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chen Z, Tyler DE (2002) The influence function and maximum bias of Tukey’s median. Ann Stat 30:1737–1759MathSciNetCrossRefzbMATHGoogle Scholar
  4. Davies PL (1990) The asymptotics of S-estimators in the linear regression model. Ann Stat 18:1651–1675MathSciNetCrossRefzbMATHGoogle Scholar
  5. Davies PL (1993) Aspects of robust linear regression. Ann Stat 21:1843–1899MathSciNetCrossRefzbMATHGoogle Scholar
  6. Davies PL, Gather U (2005) Breakdown and groups. Ann Stat 33(3):977–988MathSciNetCrossRefzbMATHGoogle Scholar
  7. Donoho DL (1982) Breakdown properties of multivariate location estimators. PhD Qualifying Paper, Harvard UniversityGoogle Scholar
  8. Donoho DL, Huber P (1983) A Festschrift for Erich L. Lehmann. Wadsworth, Belmont, pp 157–184Google Scholar
  9. Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA (1986) Robust statistics: the approach based on influence functions. Wiley, New YorkzbMATHGoogle Scholar
  10. Huber PJ (1964) Robust estimation of a location parameter. Ann Math Stat 35:73–101MathSciNetCrossRefzbMATHGoogle Scholar
  11. Huber PJ (1972) Robust statistics: a review. Ann Math Stat 43:1041–1067CrossRefzbMATHGoogle Scholar
  12. Huber PJ (1981) Robust statistics. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  13. Hubert M, Rousseeuw PJ, Van Aelst S (2001) Similarities between location depth and regression depth. In: Fernholz L, Morgenthaler S, Stahel W (eds) Statistics in genetics and in the environmental sciences. Birkhäuser, Basel, pp 159–172CrossRefGoogle Scholar
  14. Kim J, Pollard D (1990) Cube root asymptotics. Ann Stat 18:191–219MathSciNetCrossRefzbMATHGoogle Scholar
  15. Koenker R, Bassett GJ (1978) Regression quantiles. Econometrica 46:33–50MathSciNetCrossRefzbMATHGoogle Scholar
  16. Liu X, Luo S, Zuo Y (2017) Some results on the computing of Tukey’s halfspace median. Stat Pap. Google Scholar
  17. Maronna RA, Yohai VJ (1993) Bias-robust estimates of regression based on projections. Ann Stat 21(2):965–990MathSciNetCrossRefzbMATHGoogle Scholar
  18. Martin DR, Yohai VJ, Zamar RH (1989) Min–max bias robust regression. Ann Stat 17:1608–1630MathSciNetCrossRefzbMATHGoogle Scholar
  19. Müller C (2013) Upper and lower bounds for breakdown points. In: Becker C, Fried R, Kuhnt S (eds) Robustness and complex data structures. Festschrift in Honour of Ursula Gather. Springer, Berlin, pp 17–34Google Scholar
  20. Rousseeuw PJ (1984) Least median of squares regression. J Am Stat Assoc 79:871–880MathSciNetCrossRefzbMATHGoogle Scholar
  21. Rousseeuw PJ, Hubert M (1999) Regression depth (with discussion). J Am Stat Assoc 94:388–433CrossRefzbMATHGoogle Scholar
  22. Rousseeuw PJ, Leroy A (1987) Robust regression and outlier detection. Wiley, New York 1987CrossRefzbMATHGoogle Scholar
  23. Seber GAF, Lee AJ (2003) Linear regression analysis, 2nd edn. Wiley, Hoboken, NJCrossRefzbMATHGoogle Scholar
  24. Shao W, Zuo Y (2019) Computing the halfspace depth with multiple try algorithm and simulated annealing algorithm. Comput Stat.
  25. Tukey JW (1975) Mathematics and the picturing of data. In: James RD (ed) Proceeding of the international congress of mathematicians, Vancouver 1974, vol 2. Canadian Mathematical Congress, Montreal, pp 523–531Google Scholar
  26. Van Aelst S, Rousseeuw PJ (2000) Robustness of deepest regression. J Multivar Anal 73:82–106MathSciNetCrossRefzbMATHGoogle Scholar
  27. Wu M, Zuo Y (2008) Trimmed and Winsorized standard deviations based on a scaled deviation. J Nonparametric Stat 20(4):319–335MathSciNetCrossRefzbMATHGoogle Scholar
  28. Wu M, Zuo Y (2009) Trimmed and Winsorized means based on a scaled deviation. J Stat Plan Inference 139(2):350–365MathSciNetCrossRefzbMATHGoogle Scholar
  29. Zuo Y (2003) Projection-based depth functions and associated medians. Ann Stat 31:1460–1490MathSciNetCrossRefzbMATHGoogle Scholar
  30. Zuo Y (2006) Multi-dimensional trimming based on projection depth. Ann Stat 34(5):2211–2251CrossRefzbMATHGoogle Scholar
  31. Zuo Y, Cui H, He X (2004) On the Stahel–Donoho estimator and depth-weighted means of multivariate data. Ann Stat 32(1):167–188MathSciNetCrossRefzbMATHGoogle Scholar
  32. Zuo Y, Cui H, Young D (2004) Influence function and maximum bias of projection depth based estimators. Ann Stat 32:189–218MathSciNetCrossRefzbMATHGoogle Scholar
  33. Zuo Y (2018) On general notions of depth in regression. arXiv:1805.02046

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA

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