Robustness to Outliers of Bounded-Error Estimators and Consequences on Experiment Design

  • L. Pronzato
  • É. Walter

Abstract

If proper precautions are not taken, bounded-error estimators are not robust to outliers, i.e., to data points where the actual error is larger than assumed when specifying the error bounds. The outlier minimal number estimator (OMNE) has been designed to overcome this difficulty and has proved on various examples to be particularly insensitive to outliers. This chapter is devoted to a theoretical study of its robustness. The notion of breakdown point, introduced to quantify the robustness of point estimators, is extended to set-estimators. When the model output is linear in the parameters, OMNE is shown to possess the highest achievable breakdown point. A bound on the bias due to outliers is established and used to define a new policy for optimal experimental design aimed at providing a higher protection against outliers than conventional D-optimal design.

Keywords

Design Matrix Full Rank Design Policy Breakdown Point Regular Data 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    R. L. Launer and G. N. Wilkinson, editors, Robustness in Statistics. Academic Press, New York (1979).MATHGoogle Scholar
  2. 2.
    P. J. Rousseeuw and A. M. Leroy, Robust Regression and Outlier Detection. Wiley, New York (1987).MATHCrossRefGoogle Scholar
  3. 3.
    J. P. Norton, Automatica 23, 497 (1987).MATHCrossRefGoogle Scholar
  4. 4.
    J. R. Deller, IEEE ASSP Mag. 6, 4 (1989).CrossRefGoogle Scholar
  5. 5.
    M. Milanese, in: Robustness in Identification and Control (M. Milanese, R. Tempo, and A. Vicino, eds.) Plenum Press, New York pp. 3–24 (1989).CrossRefGoogle Scholar
  6. 6.
    E. Walter and H. Piet-Lahanier, Math. Comput. Simul. 32, 449 (1990).MathSciNetCrossRefGoogle Scholar
  7. 7.
    E. Walter and H. Piet-Lahanier, in: Proceedings of the 25th IEEE Conference on Decision and Control, Athens, Greece pp. 1037-1042 (1986).Google Scholar
  8. 8.
    H. Lahanier, E. Walter, and R. Gomeni, J. Pharm. Biopharm. 15, 203 (1987).Google Scholar
  9. 9.
    E. Walter and H. Piet-Lahanier, in: Robustness in Identification and Control (M. Milanese, R. Tempo, and A. Vicino, eds.) Plenum Press, New York pp. 67–76 (1989).CrossRefGoogle Scholar
  10. 10.
    J. P. Norton, in: Proceedings of the 25th IEEE Conference on Decision and Control. Athens, Greece pp. 286-290 (1986).Google Scholar
  11. 11.
    D. P. Bertsekas and I. B. Rhodes, IEEE Trans. Autom. Control 16, 117 (1971).MathSciNetCrossRefGoogle Scholar
  12. 12.
    E. Walter and H. Piet-Lahanier, IEEE Trans. Autom. Control 34, 911 (1989).MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    L. Pronzato and E. Walter, in: Optimal Design and Analysis of Experiments (Y. Dodge, V. V. Federov, and H. P. Wynn, eds.) North-Holland, Amsterdam, The Netherlands, pp. 195–205 (1988).Google Scholar
  14. 14.
    L. Pronzato and E. Walter, Automatica 25, 383 (1989).MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    E. Walter and H. Piet-Lahanier, Math. Biosci. 92, 55 (1988).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Ch. H. Müller, J. Stat. Plan. Inf. (1995) in press.Google Scholar
  17. 17.
    G. E. P. Box and N. R. Draper, Biometrika 62, 347 (1975).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • L. Pronzato
    • 1
  • É. Walter
    • 2
  1. 1.Laboratoire I3S, CNRS URA-1376, Sophia AntipolisValbonneFrance
  2. 2.Laboratoire des Signaux et SystèmesCNRS-École Supérieure d’ElectricitéGif-sur-Yvette CedexFrance

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