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On the relation between S-estimators and M-estimators of multivariate location and covariance. (English) Zbl 0702.62031
M-estimators of multivariate location and covariance based on $$\underset{\tilde{}} x_ 1,...,\underset{\tilde{}} x_ n\in {\mathbb{R}}^ p$$ are considered as defined by P. J. Huber [Robust statistics. (1981; Zbl 0536.62025)]. For the same problem, the S- estimator based on $$\rho:\;{\mathbb{R}}\to [0,\infty)$$ and satisfying certain conditions, is defined as the solution $${\underset{\tilde{}} \theta}_ n=(\underset{\tilde{}} t_ n,\underset{\tilde{}} C_ n)$$ to the problem of minimizing det($$\underset{\tilde{}} C)$$ subject to $n^{-1}\sum^{n}_{i=1}\rho [\{(\underset{\tilde{}} x_ i- \underset{\tilde{}} t)^ T\underset{\tilde{}} C^{- 1}(\underset{\tilde{}} x_ i-\underset{\tilde{}} t)\}^{1/2}]=b_ 0,$ for a certain constant $$b_ 0$$. S-estimators are shown to satisfy the first-order conditions of M-estimators, as is the case in the estimation in multiple regression where S-estimators were originally proposed [see P. Rousseeuw and V. Yohai, Robust and nonlinear time series analysis, Proc. Workshop, Heidelberg/Ger. 1983, Lect. Notes Stat. 26, 256-272 (1984; Zbl 0567.62027)].
It is shown that the influence function of S-functionals exists and is the same as that of the corresponding M-functionals, and also the S- estimators have a limiting normal distribution which is similar to the limiting normal distribution of M-estimators. Finally the author concludes that the S-estimator is able to achieve the asymptotic variances attained by the M-estimator, but in addition it has a breakdown point that becomes considerably higher when the dimension p increases.
Reviewer: R.Mentz

##### MSC:
 62F35 Robustness and adaptive procedures (parametric inference) 62H12 Estimation in multivariate analysis 62E20 Asymptotic distribution theory in statistics
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