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Asymptotics of Generalized S-Estimators

Author

Listed:
  • Hossjer, O.
  • Croux, C.
  • Rousseeuw, P. J.

Abstract

An S-estimator of regression is obtained by minimizing an M-estimator of scale applied to the residuals ri. On the other hand, a generalized S-estimator (or GS-estimator) minimizes an M-estimator of scale based on all pairwise differences ri - rj. Generalized S-estimators have similar robustness properties as S-estimators, including a high breakdown point. In this paper we prove asymptotic normality for the GS-esimator of the regression parameters, as well as for the accompanying scale estimator defined by the minimal value of the objective function. It turns out that the asymptotic efficiency can be much higher than that of S-estimators. For instance, by using a biweight [rho]-function we obtain a GS-estimator with 50% breakdown point and 68.4% efficiency.

Suggested Citation

  • Hossjer, O. & Croux, C. & Rousseeuw, P. J., 1994. "Asymptotics of Generalized S-Estimators," Journal of Multivariate Analysis, Elsevier, vol. 51(1), pages 148-177, October.
    • Handle: RePEc:eee:jmvana:v:51:y:1994:i:1:p:148-177
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    Cited by:

    1. Roelant, E. & Van Aelst, S. & Croux, C., 2009. "Multivariate generalized S-estimators," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 876-887, May.
    2. Jun, Sung Jae & Pinkse, Joris & Wan, Yuanyuan, 2011. "-Consistent robust integration-based estimation," Journal of Multivariate Analysis, Elsevier, vol. 102(4), pages 828-846, April.
    3. Ma, Yanyuan & Genton, Marc G., 2001. "Highly Robust Estimation of Dispersion Matrices," Journal of Multivariate Analysis, Elsevier, vol. 78(1), pages 11-36, July.
    4. Farebrother, Richard William, 1997. "The historical development of the linear minimax absolute residual estimation procedure 1786-1960," Computational Statistics & Data Analysis, Elsevier, vol. 24(4), pages 455-466, June.
    5. Hawkins, Douglas M. & Olive, David, 1999. "Applications and algorithms for least trimmed sum of absolute deviations regression," Computational Statistics & Data Analysis, Elsevier, vol. 32(2), pages 119-134, December.
    6. Cizek, P., 2009. "Generalized Methods of Trimmed Moments," Discussion Paper 2009-25, Tilburg University, Center for Economic Research.
    7. Nunkesser, Robin & Morell, Oliver, 2008. "Evolutionary algorithms for robust methods," Technical Reports 2008,29, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    8. Agulló, Jose & Croux, Christophe & Van Aelst, Stefan, 2008. "The multivariate least-trimmed squares estimator," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 311-338, March.
    9. Bernholt, Thorsten & Nunkesser, Robin & Schettlinger, Karen, 2007. "Computing the least quartile difference estimator in the plane," Computational Statistics & Data Analysis, Elsevier, vol. 52(2), pages 763-772, October.
    10. Berrendero, José R. & Zamar, Ruben H., 1999. "Global robustness of location and dispersion estimates," Statistics & Probability Letters, Elsevier, vol. 44(1), pages 63-72, August.
    11. Cerioli, Andrea & Farcomeni, Alessio & Riani, Marco, 2014. "Strong consistency and robustness of the Forward Search estimator of multivariate location and scatter," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 167-183.
    12. Berrendero, José R., 2003. "Uniform strong consistency of robust estimators," Statistics & Probability Letters, Elsevier, vol. 64(2), pages 159-168, August.
    13. Ian L. Dryden & Gary Walker, 1999. "Highly Resistant Regression and Object Matching," Biometrics, The International Biometric Society, vol. 55(3), pages 820-825, September.
    14. W. Ip & Ying Yang & P. Kwan & Y. Kwan, 2003. "Strong convergence rate of the least median absolute estimator in linear regression models," Statistical Papers, Springer, vol. 44(2), pages 183-201, April.
    15. Farnè, Matteo & Vouldis, Angelos T., 2018. "A methodology for automised outlier detection in high-dimensional datasets: an application to euro area banks' supervisory data," Working Paper Series 2171, European Central Bank.
    16. Nunkesser, Robin & Morell, Oliver, 2010. "An evolutionary algorithm for robust regression," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3242-3248, December.
    17. Croux, Christophe & Ruiz-Gazen, Anne, 2005. "High breakdown estimators for principal components: the projection-pursuit approach revisited," Journal of Multivariate Analysis, Elsevier, vol. 95(1), pages 206-226, July.
    18. Kudraszow, Nadia L. & Maronna, Ricardo A., 2011. "Estimates of MM type for the multivariate linear model," Journal of Multivariate Analysis, Elsevier, vol. 102(9), pages 1280-1292, October.
    19. Bernholt, Thorsten & Nunkesser, Robin & Schettlinger, Karen, 2005. "Computing the Least Quartile Difference Estimator in the Plane," Technical Reports 2005,51, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    20. Agullo, Jose, 2001. "New algorithms for computing the least trimmed squares regression estimator," Computational Statistics & Data Analysis, Elsevier, vol. 36(4), pages 425-439, June.
    21. Sirkiä, Seija & Taskinen, Sara & Oja, Hannu, 2007. "Symmetrised M-estimators of multivariate scatter," Journal of Multivariate Analysis, Elsevier, vol. 98(8), pages 1611-1629, September.
    22. Soukissian, Takvor H. & Karathanasi, Flora E., 2016. "On the use of robust regression methods in wind speed assessment," Renewable Energy, Elsevier, vol. 99(C), pages 1287-1298.

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