Domains of weak continuity of statistical functionals with a view toward robust statistics

In Kr\"atschmer/Schied/Z\"ahle (2012) and Kr\"atschmer/Schied/Z\"ahle (2014) a refined notion of (qualitative) robustness has been suggested which applies to tail-dependent statistical functionals and allows to compare different statistical functionals in regard to their degree of robustness. Recently, it was observed in Z\"ahle (2016) that this new concept may be viewed as a localization of the classical concept by Hampel. By generalizing Hampel's theorem, local robustness for the weak topology may be linked with continuity of statistical functionals for topologies on sets of distributions which are finer than the weak topology. So the crucial point in Z\"ahle (2016) was to identify those sets on which the finer topologies coincide with the weak topologies. In the present paper the generalized Hampel theorem is extended to larger classes of topologies on sets of distributions, and several characterizations of those sets on which the finer topologies coincide with the weak topologies are given. As an application robustness of maximum likelihood estimators is studied. A further application deals with robustness of the empirical estimators of law-invariant convex risk measures, complementing investigations of Kr\"atschmer/Schied/Z\"ahle (2014).