Skip to main content
Log in

Computing least trimmed squares regression with the forward search

  • Published:
Statistics and Computing Aims and scope Submit manuscript

Abstract

Least trimmed squares (LTS) provides a parametric family of high breakdown estimators in regression with better asymptotic properties than least median of squares (LMS) estimators. We adapt the forward search algorithm of Atkinson (1994) to LTS and provide methods for determining the amount of data to be trimmed. We examine the efficiency of different trimming proportions by simulation and demonstrate the increasing efficiency of parameter estimation as larger proportions of data are fitted using the LTS criterion. Some standard data examples are analysed. One shows that LTS provides more stable solutions than LMS.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Atkinson, A. C. (1985) Plots, Transformations and Regression, Oxford: Oxford University Press.

    Google Scholar 

  • Atkinson, A. C. (1994) Fast very robust methods for the detection of multiple outliers. Journal of the American Statistical Association, 89, 1329–1339.

    Google Scholar 

  • Atkinson, A. C. and Mulira, H.-M. (1993) The stalactite plot for the detection of multivariate outliers. Statistics and Computing, 3, 27–35.

    Google Scholar 

  • Coakley, C. W., Mili, L. and Cheniae, M. G. (1994) Effect of leverage on the finite sample effciencies of high breakdown estimators. Statistics and Probability Letters, 19, 399–408.

    Google Scholar 

  • Croux, C., Rousseeuw, P. J. and Hössjer, O. (1994) Generalized S-estimators. Journal of the American Statistical Association, 89, 1271–1281.

    Google Scholar 

  • Davies, L. (1994) Desirable properties, breakdown and effciency in the linear regression model. Statistics and Probability Letters, 19, 361–370.

    Google Scholar 

  • Davies, P. L. (1993) Aspects of robust linear regression. The Annals of Statistics, 21, 1843–1899.

    Google Scholar 

  • Hadi, A. S. and Luceño, A. (1997) Maximum trimmed likelihood estimators: a unified approach, examples, and algorithms. Computational Statistics and Data Analysis, 25, 251–272.

    Google Scholar 

  • Hawkins, D. M. (1994) The feasible solution algorithm for least trimmed squares regression. Computational Statistics and Data Analysis, 17, 185–196.

    Google Scholar 

  • Hettmansperger, T. P. and McKean, J. W. (1998) Robust Nonparametric Statistical Methods, London: Arnold.

    Google Scholar 

  • Hettmansperger, T. P. and Sheather, S. J. (1992) A cautionary note on the method of the least median squares. The American Statistician, 46, 79–83.

    Google Scholar 

  • Hettmansperger, T. P. and Sheather, S. J. (1993) Reply to the comment of Rousseeuw, P. J., The American Statistician, 47, 162–163.

    Google Scholar 

  • Hössjer, O. (1994) Rank-based estimates in the linear model With high breakdown point. Journal of the American Statistical Association, 89, 149–157.

    Google Scholar 

  • Hössjer, O. (1995) Exact computation of the least trimmed squares estimate in simple linear regression. Computational Statistics and Data Analysis, 19, 265–282.

    Google Scholar 

  • Morgenthaler, S. (1991) A note on effcient regression estimators with positive breakdown point. Statistics and Probability Letters, 11, 469–472.

    Google Scholar 

  • Rousseeuw, P. J. (1984) Least median of squares regression. Journal of the American Statistical Association, 79, 871–880.

    Google Scholar 

  • Rousseeuw, P. J. (1993) Comment on the paper of Hettmansperger, T. P. and Sheather, S. J., The American Statistician, 47, 162.

    Google Scholar 

  • Rousseeuw, P. J. (1994) Unconventional features of positivebreakdown estimators. Statistics and Probability Letters, 19, 417–431.

    Google Scholar 

  • Rousseeuw, P. J. and Leroy, A. M. (1987) Robust Regression and Outlier Detection, New York: John Wiley.

    Google Scholar 

  • Stefanski, L. A. (1991) A note on high breakdown point estimators. Statistics and Probability Letters, 11, 353–358.

    Google Scholar 

  • Yohai, V. J. (1987) High breakdown point and high effciency robust estimators for regression. The Annals of Statistics, 15, 462–656.

    Google Scholar 

  • Yohai, V. J. and Zamar, R. H. (1988) High breakdown point estimates of regression by means of the minimization of an effcient scale. Journal of the American Statistical Association, 83, 406–413.

    Google Scholar 

Download references

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Atkinson, A.C., Cheng, TC. Computing least trimmed squares regression with the forward search. Statistics and Computing 9, 251–263 (1999). https://doi.org/10.1023/A:1008942604045

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008942604045

Navigation