## On the optimality of multivariate S-estimators.(English)Zbl 1246.62136

Summary: We maximize the efficiency of a multivariate $$S$$-estimator under a constraint on the breakdown point. In the linear regression model, it is known that the highest possible efficiency of a maximum breakdown $$S$$-estimator is bounded above by 33 per cent for Gaussian errors. We prove the surprising result that in dimensions larger than one, the efficiency of a maximum breakdown $$S$$-estimator of location and scatter can get arbitrarily close to 100 per cent, by an appropriate selection of the loss function.

### MSC:

 62H12 Estimation in multivariate analysis 62F35 Robustness and adaptive procedures (parametric inference)

robustbase
Full Text:

### References:

 [1] Bashir, High breakdown mixture discriminant analysis, J. Multivariate Anal. 93 pp 102– (2005) · Zbl 1087.62076 [2] Croux, Efficient high-breakdown M-estimators of scale, Statist. Probab. Lett. 19 pp 371– (1994) · Zbl 0791.62034 [3] Croux, Robust linear discriminant analysis using S-estimators, Canad. J. Statist. 29 pp 473– (2001) · Zbl 0987.62044 [4] Croux, Principal component analysis based on robust estimators of the covariance or correlation matrix: influence functions and efficiencies, Biometrika 87 pp 603– (2000) · Zbl 0956.62047 [5] Croux , C. Dehon , C. Yadine , A. 2009 Technical note: on the optimality of multivariate S-estimators http://www.econ.kuleuven.be/christophe.croux/public/public.htm · Zbl 1284.62367 [6] Davies, Asymptotic behaviour of S-estimates of multivariate location parameters and dispersion matrices, Ann. Statist. 15 pp 1269– (1987) · Zbl 0645.62057 [7] Frahm, Asymptotic distributions of robust shape matrices and scale, J. Multivariate Anal. 100 pp 1329– (2009) · Zbl 1274.62130 [8] Hampel, Robust statistics: the approach based on influence functions (1986) · Zbl 0593.62027 [9] Hössjer, On the optimality of S-estimators, Statist. Probab. Lett. 14 pp 413– (1992) · Zbl 0761.62036 [10] Lopuhaä, On the relation between S-estimators and M-estimators of multivariate location and covariance, Ann. Statist. 17 pp 1662– (1989) · Zbl 0702.62031 [11] Maronna, Robust statistics: theory and methods (2006) · Zbl 1094.62040 [12] Salibian-Barrera, A fast algorithm for S-regression estimates, J. Comput. Graph. Statist. 15 pp 414– (2006) · Zbl 1090.62029 [13] Tatsuoka, On the uniqueness of S-functionals and M-functionals under nonelliptical distributions, Ann. Statist. 28 pp 1219– (2000) · Zbl 1105.62347 [14] Yohai, High breakdown-point and high-efficiency robust estimates for regression, Ann. Statist. 15 pp 642– (1987) · Zbl 0624.62037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.