The "automatic" robustness of minimum distance functionals

The minimum distance (MD) functional defined by a distance μμ is automatically robust over contamination neighborhoods defined by μμ. In fact, when compared to other Fisher-consistent functionals, the MD functional was no worse than twice the minimum sensitivity to μμ-contamination, and at least half the best possible breakdown point. In invariant settings, the MD functional has the best attainable breakdown point against μμ-contamination among equivariant functionals. If μμ is Hilbertian (e.g., the Hellinger distance), the MD functional has the smallest sensitivity to μμ-contamination among Fisher-consistent functionals. The robustness of the MD functional is inherited by MD estimates, both estimates based on "weak" distances and estimates based on "strong" distances, when the empirical distribution is appropriately smoothed. These facts are general and apply not just in simple location models, but also in multivariate location-scatter and in semiparametric settings. Of course, this robustness is formal because μμ-contamination neighborhoods may not be large enough to contain realistic departures from the model. For the metrics we are interested in, robustness against μμ-contamination is stronger than robustness against gross errors contamination; and for "weak" metrics (e.g., μ=Cramer-von Mises, Kolmogorov)μ=Cramer-von Mises, Kolmogorov), robustness over μμ-neighborhoods implies robustness over Prohorov neighborhoods.