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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

Citations:

Zbl 0609.62054
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References:

[1] Clarke, B.R., Uniqueness and Fréchet differentiability of functional solutions to maximum likelihood type equations, Ann. státist., 4, 1196-1205, (1983) · Zbl 0541.62023
[2] Davies, P.L., Asymptotic behaviour of S-estimates of multivariate location parameters and dispersion matrices, Ann. statist., 15, 1269-1292, (1987) · Zbl 0645.62057
[3] Davies, P.L., The asymptotics of Rousseeuw’s minimum volume ellipsoid estimator, (1990), University of Essen, Preprint
[4] Hampel, F.R.; Ronchetti, E.M.; Rousseeuw, P.J.; Stahel, W.A., ()
[5] Huber, P.J., ()
[6] Jureckova, J.; Portnoy, S., Asymptotics for one-step M-estimators in regression with application to combining efficiency and high breakdown point, Commun. statist. theory meth., 16, No. 8, 2187-2199, (1987) · Zbl 0656.62041
[7] Kiefer, J.C., ()
[8] Lopuhaä, H.P., On the relation between S-estimators and M-estimators of multivariate location and covariance, Ann. statist., 17, 1662-1683, (1989) · Zbl 0702.62031
[9] Lopuhaä, H.P., Multivariate τ-estimators for location and scatter, (1990), University of Delft, Preprint · Zbl 0746.62034
[10] Lopuhaä, H.P.; Rousseeuw, P.J., Breakdown points of affine equivariant estimators of multivariate location and covariance matrices, Ann. statist., 19, 229-248, (1991) · Zbl 0733.62058
[11] Rousseeuw, P.J., Least Median of squares regression, J. am. statist. assoc., 79, 871-880, (1984) · Zbl 0547.62046
[12] Rousseeuw, P.J., Multivariate estimation with high breakdown point, (), 288-297
[13] Rousseeuw, P.J.; Yohai, V., Robust regression by means of S-estimators, (), 256-272 · Zbl 0567.62027
[14] Uhrmann, E., Ein-schrift M-schätzer für multivariaten mittelwert und varianz-kovarianz-matrix, ()
[15] Yohai, V.J., High breakdown-point and high efficiency robust estimates for regression, Ann. statist., 15, 642-656, (1987) · Zbl 0624.62037
[16] Yohai, V.J.; Zamar, R.H., High breakdown-point estimates of regression by means of the minimization of an efficient scale, J. am. statist. assoc., 83, No. 402, 406-413, (1988) · Zbl 0648.62036
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