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The Deepest Regression Method

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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.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Multivariate Analysis.

Volume (Year): 81 (2002)
Issue (Month): 1 (April)
Pages: 138-166

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Handle: RePEc:eee:jmvana:v:81:y:2002:i:1:p:138-166

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Related research

Keywords: regression depth algorithm inference;

References

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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 1115-23, 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. 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.
  2. 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.
  3. 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.
  4. 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.
  5. Chernozhukov, Victor & Hong, Han, 2003. "An MCMC approach to classical estimation," Journal of Econometrics, Elsevier, vol. 115(2), pages 293-346, August.
  6. Ursula Gather & Karen Schettlinger & Roland Fried, 2006. "Online signal extraction by robust linear regression," Computational Statistics, Springer, vol. 21(1), pages 33-51, March.
  7. 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.
  8. Christine Müller, 2011. "Data depth for simple orthogonal regression with application to crack orientation," Metrika, Springer, vol. 74(2), pages 135-165, September.
  9. Stephan Morgenthaler, 2007. "A survey of robust statistics," Statistical Methods and Applications, Springer, vol. 15(3), pages 271-293, February.
  10. 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.
  11. 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.
  12. Wellmann, Robin & Müller, Christine H., 2010. "Depth notions for orthogonal regression," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2358-2371, November.

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