The "Automatic" Robustness of Minimum Distance Functionals

Abstract

The minimum distance (MD) functional defined by a distance $\mu$ is automatically robust over contamination neighborhoods defined by $\mu$. In fact, when compared to other Fisher-consistent functionals, the MD functional was no worse than twice the minimum sensitivity to $\mu$-contamination, and at least half the best possible breakdown point. In invariant settings, the MD functional has the best attainable breakdown point against $\mu$-contamination among equivariant functionals. If $\mu$ is Hilbertian (e.g., the Hellinger distance), the MD functional has the smallest sensitivity to $\mu$-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 $\mu$-contamination neighborhoods may not be large enough to contain realistic departures from the model. For the metrics we are interested in, robustness against $\mu$-contamination is stronger than robustness against gross errors contamination; and for "weak" metrics (e.g., $\mu = \text{Cramer-von Mises, Kolmogorov})$, robustness over $\mu$-neighborhoods implies robustness over Prohorov neighborhoods.