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An efficient Fréchet differentiable high breakdown multivariate location and dispersion estimator

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  • Davies, Laurie

Abstract

A good robust functional should, if possible, be efficient at the model, smooth, and have a high breakdown point. M-estimators can be made efficient and Fréchet differentiable by choosing appropriate [psi]-functions but they have a breakdown point of at most 1/(p + 1) in p dimensions. On the other hand, the local smoothness of known high breakdown functionals has not been investigated. It is known that Rousseeuw's minimum volume ellipsoid estimator is not differentiable and that S-estimators based on smooth functions force a trade-off between efficiency and breakdown point. However, by using a two-step M-estimator based on the minimum volume ellipsoid we show that it is possible to obtain a highly efficient, Fréchet differentiable estimator whilst still retaining the breakdown point. This result is extended to smooth S-estimators.

Suggested Citation

  • Davies, Laurie, 1992. "An efficient Fréchet differentiable high breakdown multivariate location and dispersion estimator," Journal of Multivariate Analysis, Elsevier, vol. 40(2), pages 311-327, February.
    • Handle: RePEc:eee:jmvana:v:40:y:1992:i:2:p:311-327
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    Cited by:

    1. Croux, Christophe & Haesbroeck, Gentiane, 1999. "Influence Function and Efficiency of the Minimum Covariance Determinant Scatter Matrix Estimator," Journal of Multivariate Analysis, Elsevier, vol. 71(2), pages 161-190, November.
    2. Bianco, Ana & Boente, Graciela, 2002. "On the asymptotic behavior of one-step estimates in heteroscedastic regression models," Statistics & Probability Letters, Elsevier, vol. 60(1), pages 33-47, November.

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