## An efficient Fréchet differentiable high breakdown multivariate location and dispersion estimator.(English)Zbl 0748.62030

Summary: 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 P. J. Rousseeuw’s minimum volume ellipsoid estimator [Mathematical statistics and applications, Proc. 4th Pannonian Symp. Math. Stat., Bad Tatzmannsdorf/Austria 1983, Vol. B, 283-297 (1985; Zbl 0609.62054)] 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.

### MSC:

 62H12 Estimation in multivariate analysis 62F35 Robustness and adaptive procedures (parametric inference) 62F12 Asymptotic properties of parametric estimators

Zbl 0609.62054
Full Text:

### References:

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