Asymptotic Robustness for Shrinking Contamination
In this chapter we show that the robustness concept based on influence functions (see Section 3.2) is closely connected with a robustness concept based on shrinking neighbourhoods. In particular, the concepts coincide for estimators given by Frechet differentiable functionals as is shown in Section 5.1. But Section 5.1 also shows that for robustness concepts based on shrinking neighbourhoods also the larger class of asymptotically linear estimators can be used instead of the class of estimators which satisfy some differentiability condition. Therefore the robustness properties are given for asymptotically linear estimators. Especially, these properties are derived in Section 5.2 for estimating a linear aspect in a linear model, in Section 5.3 for estimating a nonlinear aspect in a linear model and in Section 5.4 for estimation in a nonlinear model.
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