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The Asymptotics of MM-Estimators for Linear Regression with Fixed Designs

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Abstract

MM-estimators achieve simultaneous high efficiency and high breakdown point over contamination neighborhoods. Inference based on these estimators relies on their asymptotic properties, which have been studied for the case of random covariates. In this paper we show that, under relatively mild regularity conditions, MM-estimators for linear regression models are strongly consistent when the design is fixed. Moreover, their strong consistency allows us to show that these estimators are also asymptotically normal for non-random covariates. These results justify the use of a normal approximation to the finite-sample distribution of MM-estimators for linear regression with fixed explanatory variables. Additionally, these results have been used to extend the robust bootstrap (Salibian-Barrera and Zamar in Ann Stat 30:556–582, 2002) to the case of fixed designs [see Salibian-Barrera 2004, submitted].

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References

  • Beaton AE, Tukey JW (1974) The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data. Technometrics 16:147–185

    Article  MATH  Google Scholar 

  • Davies PL (1990) The asymptotics of S-estimators in the linear regression model. Ann Stat 18:1651–1675

    Article  MATH  Google Scholar 

  • Donoho DL, Huber PJ (1983). The notion of breakdown-point. In: Bickel PJ, Doksum KA, Hodges JL Jr (eds). A festschrift for Erich L. Lehmann. Wadsworth, Belmont California, pp 157–184

    Google Scholar 

  • Hampel FR (1971) A general qualitative definition of robustness. Ann Math Stat 42:1887–1896

    Article  MathSciNet  MATH  Google Scholar 

  • He X, Shao Q (1996) A general Bahadur representation of M-estimators and its application to linear regression with non-stochastic designs. Ann Stat 24:2608–2630

    Article  MathSciNet  MATH  Google Scholar 

  • Hössjer O (1994) Rank-based estimates in the linear model with high-breakdown point. J Am Stat Assoc 89:149–158

    Article  MATH  Google Scholar 

  • Kim J, Pollard D (1990) Cube root asymptotics. Ann Stat 18:191–219

    Article  MathSciNet  MATH  Google Scholar 

  • Martinsek AT (1989) Almost sure expansions for M-estimators and S-estimators in regression. Technical report # 25, Department of Statistics, University of Illinois

  • Rousseeuw PJ (1984) Least median of squares regression. J Am Stat Assoc 79:871–880

    Article  MathSciNet  MATH  Google Scholar 

  • Rousseeuw PJ, Leroy AM (1987) Robust regression and outlier detection. Wiley, New York

    MATH  Google Scholar 

  • Rousseeuw PJ, Yohai VJ (1984). Robust regression by means of S-estimators. In: Franke J, Hardle W, Martin RD (eds). Robust and nonlinear time series. lecture notes in statistics no. 26. Springer, Berlin Heidelberg New York, pp 256–272

    Google Scholar 

  • Salibian-Barrera, M (2004) Bootstrapping MM-estimators for linear regression with fixed designs (Submitted) Available on-line: at http://hajek.stat.ubc.ca/matias/pubs.html

  • Salibian-Barrera M, Zamar RH (2002) Bootstrapping robust estimates of regression. Ann Stat 30:556–582

    Article  MathSciNet  MATH  Google Scholar 

  • Silvapulle MJ (1985) Asymptotic behavior of robust estimators of regression and scale parameters with fixed carriers. Ann Stat 13:1490–1497

    Article  MathSciNet  MATH  Google Scholar 

  • Tableman M (1994) The influence functions for the least trimmed squares and the least trimmed absolute deviations estimators. Stat Prob Lett 19:329–337

    Article  MathSciNet  MATH  Google Scholar 

  • Wiens DP (1996) Asymptotics of generalized M-estimation of regression and scale with fixed carriers, in an approximately linear model. Stat Probab Lett 30:271–285

    Article  MathSciNet  MATH  Google Scholar 

  • Yohai VJ (1987) High breakdown-point and high efficiency robust estimates for regression. Ann Stat 15:642–656

    Article  MathSciNet  MATH  Google Scholar 

  • Yohai VJ, Maronna RA (1976) Location estimators based on linear combinations of modified order statistics. Commun Stat Theory Methods 5:481–486

    Article  MathSciNet  Google Scholar 

  • Yohai VJ, Maronna RA (1979) Asymptotic behavior of M-estimators for the linear model. Ann Stat 7:258–268

    Article  MathSciNet  MATH  Google Scholar 

  • Yohai VJ, Zamar RH (1988) High breakdown point estimates of regression by means of the minimization of an efficient scale. J Am Stat Assoc 83:406–413

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Statistics, University of British Columbia, 333 - 6356 Agricultural Road, Vancouver, BC, V6T 1Z2, Canada

    Matias Salibian-Barrera

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  1. Matias Salibian-Barrera
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Correspondence to Matias Salibian-Barrera.

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Research supported by an NSERC Research Grant (Individual)

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Salibian-Barrera, M. The Asymptotics of MM-Estimators for Linear Regression with Fixed Designs. Metrika 63, 283–294 (2006). https://doi.org/10.1007/s00184-005-0019-6

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  • DOI: https://doi.org/10.1007/s00184-005-0019-6

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