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Robust estimation for the multivariate linear model based on a [tau]-scale

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  • Ben, Marta García
  • Martínez, Elena
  • Yohai, Víctor J.

Abstract

We introduce a class of robust estimates for multivariate linear models. The regression coefficients and the covariance matrix of the errors are estimated simultaneously by minimizing the determinant of the covariance matrix estimate, subject to a constraint on a robust scale of the Mahalanobis norms of the residuals. By choosing a [tau]-estimate as a robust scale, the resulting estimates combine good robustness properties and asymptotic efficiency under Gaussian errors. These estimates are asymptotically normal and in the case where the errors have an elliptical distribution, their asymptotic covariance matrix differs only by a scalar factor from the one corresponding to the maximum likelihood estimate. We derive the influence curve and prove that the breakdown point is close to 0.5. A Monte Carlo study shows that our estimates compare favorably with respect to S-estimates.

Suggested Citation

  • Ben, Marta García & Martínez, Elena & Yohai, Víctor J., 2006. "Robust estimation for the multivariate linear model based on a [tau]-scale," Journal of Multivariate Analysis, Elsevier, vol. 97(7), pages 1600-1622, August.
    • Handle: RePEc:eee:jmvana:v:97:y:2006:i:7:p:1600-1622
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    References listed on IDEAS

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    1. Croux, Christophe, 1994. "Efficient high-breakdown M-estimators of scale," Statistics & Probability Letters, Elsevier, vol. 19(5), pages 371-379, April.
    2. Hössjer, Ola, 1992. "On the optimality of S-estimators," Statistics & Probability Letters, Elsevier, vol. 14(5), pages 413-419, July.
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    Cited by:

    1. Andrea Bergesio & María Eugenia Szretter Noste & Víctor J. Yohai, 2021. "A robust proposal of estimation for the sufficient dimension reduction problem," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(3), pages 758-783, September.
    2. Roelant, E. & Van Aelst, S. & Croux, C., 2009. "Multivariate generalized S-estimators," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 876-887, May.
    3. Muler, Nora & Yohai, V´ictor J., 2013. "Robust estimation for vector autoregressive models," Computational Statistics & Data Analysis, Elsevier, vol. 65(C), pages 68-79.
    4. Sonja Kuhnt, 2010. "Breakdown concepts for contingency tables," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 71(3), pages 281-294, May.
    5. 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.

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