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The Deepest Regression Method

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

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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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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  1. 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.
  2. Van Aelst, Stefan & Rousseeuw, Peter J., 2000. "Robustness of Deepest Regression," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 82-106, April.
  3. 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.
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