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The Deepest Regression Method

Author

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  • Van Aelst, Stefan
  • Rousseeuw, Peter J.
  • Hubert, Mia
  • Struyf, Anja

Abstract

Deepest regression (DR) is a method for linear regression introduced by P. J. Rousseeuw and M. Hubert (1999, J. Amer. Statis. Assoc.94, 388-402). The DR method is defined as the fit with largest regression depth relative to the data. In this paper we show that DR is a robust method, with breakdown value that converges almost surely to 1/3 in any dimension. We construct an approximate algorithm for fast computation of DR in more than two dimensions. From the distribution of the regression depth we derive tests for the true unknown parameters in the linear regression model. Moreover, we construct simultaneous confidence regions based on bootstrapped estimates. We also use the maximal regression depth to construct a test for linearity versus convexity/concavity. We extend regression depth and deepest regression to more general models. We apply DR to polynomial regression and show that the deepest polynomial regression has breakdown value 1/3. Finally, DR is applied to the Michaelis-Menten model of enzyme kinetics, where it resolves a long-standing ambiguity.

Suggested Citation

  • Van Aelst, Stefan & Rousseeuw, Peter J. & Hubert, Mia & Struyf, Anja, 2002. "The Deepest Regression Method," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 138-166, April.
    • Handle: RePEc:eee:jmvana:v:81:y:2002:i:1:p:138-166
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    References listed on IDEAS

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    1. Van Aelst, Stefan & Rousseeuw, Peter J., 2000. "Robustness of Deepest Regression," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 82-106, April.
    2. Benderly, Jason & Zwick, Burton, 1985. "Inflation, Real Balances, Output, and Real Stock Returns [Stock Returns, Real Activity, Inflation, and Money]," American Economic Review, American Economic Association, vol. 75(5), pages 1115-1123, December.
    3. Zuo, Yijun, 2001. "Some quantitative relationships between two types of finite sample breakdown point," Statistics & Probability Letters, Elsevier, vol. 51(4), pages 369-375, February.
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    Citations

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    Cited by:

    1. Ursula Gather & Karen Schettlinger & Roland Fried, 2006. "Online signal extraction by robust linear regression," Computational Statistics, Springer, vol. 21(1), pages 33-51, March.
    2. Chernozhukov, Victor & Hong, Han, 2003. "An MCMC approach to classical estimation," Journal of Econometrics, Elsevier, vol. 115(2), pages 293-346, August.
    3. Wellmann, Robin & Müller, Christine H., 2010. "Tests for multiple regression based on simplicial depth," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 824-838, April.
    4. Ryan Cumings-Menon, 2022. "Differentially Private Estimation via Statistical Depth," Papers 2207.12602, arXiv.org.
    5. Christine Müller, 2011. "Data depth for simple orthogonal regression with application to crack orientation," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(2), pages 135-165, September.
    6. Wellmann, Robin & Müller, Christine H., 2010. "Depth notions for orthogonal regression," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2358-2371, November.
    7. Debruyne, M. & Hubert, M. & Portnoy, S. & Vanden Branden, K., 2008. "Censored depth quantiles," Computational Statistics & Data Analysis, Elsevier, vol. 52(3), pages 1604-1614, January.
    8. Stephan Morgenthaler, 2007. "A survey of robust statistics," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 15(3), pages 271-293, February.
    9. Neykov, N.M. & Čížek, P. & Filzmoser, P. & Neytchev, P.N., 2012. "The least trimmed quantile regression," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1757-1770.
    10. Yijun Zuo, 2020. "Depth Induced Regression Medians and Uniqueness," Stats, MDPI, vol. 3(2), pages 1-13, April.
    11. Christmann, Andreas & Steinwart, Ingo & Hubert, Mia, 2006. "Robust Learning from Bites for Data Mining," Technical Reports 2006,03, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    12. Christmann, Andreas & Steinwart, Ingo & Hubert, Mia, 2007. "Robust learning from bites for data mining," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 347-361, September.
    13. Müller, Christine H., 2005. "Depth estimators and tests based on the likelihood principle with application to regression," Journal of Multivariate Analysis, Elsevier, vol. 95(1), pages 153-181, July.
    14. Zuo, Yijun, 2021. "Computation of projection regression depth and its induced median," Computational Statistics & Data Analysis, Elsevier, vol. 158(C).
    15. Stephan Morgenthaler, 2007. "A survey of robust statistics," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 15(3), pages 271-293, February.
    16. Wellmann, Robin & Harmand, Peter & Müller, Christine H., 2009. "Distribution-free tests for polynomial regression based on simplicial depth," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 622-635, April.
    17. Wellmann, R. & Katina, S. & Muller, Ch.H., 2007. "Calculation of simplicial depth estimators for polynomial regression with applications," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 5025-5040, June.

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