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Robustness of the deepest projection regression functional

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Abstract

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.

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References

  • Adrover J, Yohai VJ (2002) Projection estimates of multivariate location. Ann Stat 30:1760–1781

    Article  MathSciNet  Google Scholar 

  • Bai ZD, He X (1999) Asymptotic distributions of the maximal depth regression and multivariate location. Ann Stat 27(5):1616–1637 577-580

    Article  MathSciNet  Google Scholar 

  • Chen Z, Tyler DE (2002) The influence function and maximum bias of Tukey’s median. Ann Stat 30:1737–1759

    Article  MathSciNet  Google Scholar 

  • Davies PL (1990) The asymptotics of S-estimators in the linear regression model. Ann Stat 18:1651–1675

    Article  MathSciNet  Google Scholar 

  • Davies PL (1993) Aspects of robust linear regression. Ann Stat 21:1843–1899

    Article  MathSciNet  Google Scholar 

  • Davies PL, Gather U (2005) Breakdown and groups. Ann Stat 33(3):977–988

    Article  MathSciNet  Google Scholar 

  • Donoho DL (1982) Breakdown properties of multivariate location estimators. PhD Qualifying Paper, Harvard University

  • Donoho DL, Huber P (1983) A Festschrift for Erich L. Lehmann. Wadsworth, Belmont, pp 157–184

  • Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA (1986) Robust statistics: the approach based on influence functions. Wiley, New York

    MATH  Google Scholar 

  • Huber PJ (1964) Robust estimation of a location parameter. Ann Math Stat 35:73–101

    Article  MathSciNet  Google Scholar 

  • Huber PJ (1972) Robust statistics: a review. Ann Math Stat 43:1041–1067

    Article  Google Scholar 

  • Huber PJ (1981) Robust statistics. Wiley, New York

    Book  Google Scholar 

  • 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 159–172

    Chapter  Google Scholar 

  • Kim J, Pollard D (1990) Cube root asymptotics. Ann Stat 18:191–219

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Liu X, Luo S, Zuo Y (2017) Some results on the computing of Tukey’s halfspace median. Stat Pap. https://doi.org/10.1007/s00362-017-0941-5

    Article  MATH  Google Scholar 

  • Maronna RA, Yohai VJ (1993) Bias-robust estimates of regression based on projections. Ann Stat 21(2):965–990

    Article  MathSciNet  Google Scholar 

  • Martin DR, Yohai VJ, Zamar RH (1989) Min–max bias robust regression. Ann Stat 17:1608–1630

    Article  MathSciNet  Google Scholar 

  • 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 17–34

    Google Scholar 

  • Rousseeuw PJ (1984) Least median of squares regression. J Am Stat Assoc 79:871–880

    Article  MathSciNet  Google Scholar 

  • Rousseeuw PJ, Hubert M (1999) Regression depth (with discussion). J Am Stat Assoc 94:388–433

    Article  Google Scholar 

  • Rousseeuw PJ, Leroy A (1987) Robust regression and outlier detection. Wiley, New York 1987

    Book  Google Scholar 

  • Seber GAF, Lee AJ (2003) Linear regression analysis, 2nd edn. Wiley, Hoboken, NJ

    Book  Google Scholar 

  • Shao W, Zuo Y (2019) Computing the halfspace depth with multiple try algorithm and simulated annealing algorithm. Comput Stat. https://doi.org/10.1007/s00180-019-00906-x

  • 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 523–531

    Google Scholar 

  • Van Aelst S, Rousseeuw PJ (2000) Robustness of deepest regression. J Multivar Anal 73:82–106

    Article  MathSciNet  Google Scholar 

  • Wu M, Zuo Y (2008) Trimmed and Winsorized standard deviations based on a scaled deviation. J Nonparametric Stat 20(4):319–335

    Article  MathSciNet  Google Scholar 

  • Wu M, Zuo Y (2009) Trimmed and Winsorized means based on a scaled deviation. J Stat Plan Inference 139(2):350–365

    Article  MathSciNet  Google Scholar 

  • Zuo Y (2003) Projection-based depth functions and associated medians. Ann Stat 31:1460–1490

    Article  MathSciNet  Google Scholar 

  • Zuo Y (2006) Multi-dimensional trimming based on projection depth. Ann Stat 34(5):2211–2251

    Article  Google Scholar 

  • Zuo Y, Cui H, He X (2004) On the Stahel–Donoho estimator and depth-weighted means of multivariate data. Ann Stat 32(1):167–188

    MathSciNet  MATH  Google Scholar 

  • Zuo Y, Cui H, Young D (2004) Influence function and maximum bias of projection depth based estimators. Ann Stat 32:189–218

    MathSciNet  MATH  Google Scholar 

  • Zuo Y (2018) On general notions of depth in regression. arXiv:1805.02046

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Acknowledgements

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.

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  1. Department of Statistics and Probability, Michigan State University, East Lansing, MI, 48824, USA

    Yijun Zuo

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  1. Yijun Zuo
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Correspondence to Yijun Zuo.

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Zuo, Y. Robustness of the deepest projection regression functional. Stat Papers 62, 1167–1193 (2021). https://doi.org/10.1007/s00362-019-01129-4

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