Robust Location Estimates

Abstract

Measures of location differentiable at every density in the Hellinger metric are constructed in this paper. Differentiability entitles these location functionals to the label "robust," even though their influence curves need not be bounded and continuous. The latter properties are, in fact, associated with functionals differentiable in the Prokhorov metric. A Hellinger metric concept of minimax robustness of a location measure at a density shape $f$ is developed. Asymptotically optimal estimators are found for minimax robust location measures. Since, at $f$, their asymptotic variance equals the reciprocal of Fisher information, asymptotic efficiency at $f$ and robustness near $f$ prove compatible.