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The Catline for Deep Regression

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  • Hubert, Mia
  • Rousseeuw, Peter J.

Abstract

Motivated by the notion of regression depth (Rousseeuw and Hubert, 1996) we introduce thecatline, a new method for simple linear regression. At any bivariate data setZn={(xi, yi);i=1, ..., n} its regression depth is at leastn/3. This lower bound is attained for data lying on a convex or concave curve, whereas for perfectly linear data the catline attains a depth ofn. We construct anO(n log n) algorithm for the catline, so it can be computed fast in practice. The catline is Fisher-consistent at any linear modely=[beta]x+[alpha]+ein which the error distribution satisfies med(e  x)=0, which encompasses skewed and/or heteroscedastic errors. The breakdown value of the catline is 1/3, and its influence function is bounded. At the bivariate gaussian distribution its asymptotic relative efficiency compared to theL1line is 79.3% for the slope, and 88.9% for the intercept. The finite-sample relative efficiencies are in close agreement with these values. This combination of properties makes the catline an attractive fitting method.

Suggested Citation

  • Hubert, Mia & Rousseeuw, Peter J., 1998. "The Catline for Deep Regression," Journal of Multivariate Analysis, Elsevier, vol. 66(2), pages 270-296, August.
  • Handle: RePEc:eee:jmvana:v:66:y:1998:i:2:p:270-296
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    References listed on IDEAS

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    1. Osiewalski, Jacek & Steel, Mark F. J., 1993. "Robust bayesian inference in elliptical regression models," Journal of Econometrics, Elsevier, vol. 57(1-3), pages 345-363.
    2. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
    3. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters,in: THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78 World Scientific Publishing Co. Pte. Ltd..
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    Cited by:

    1. 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.
    2. 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.
    3. 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.

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