Before revision:  the most bias-robust estimator minimizing the maximal (asymptotic) bias and the estimatorminimizing the maximal (asymptotic) MSE. For each of these  estimators, we determine the finite sample breakdown point, the influence function, as well as  statistical accuracy measured by asymptotic bias, variance, and mean squared error - all evaluated uniformly on shrinking convex contamination neighborhoods.
After revision:  they minimize the maximal bias, respectively, on a specific such neighborhood, the maximal mean squared error. For their initialization, we propose a particular Location-Dispersion (LD) estimator, kMedMAD, which matches the population median and kMAD (an asymmetric variant of the median of absolute deviations) against the empirical counterparts. These optimally-robust estimators are compared to maximum likelihood, skipped maximum likelihood, Cramer-von-Mises minimum distance, method of median, and Pickands estimators. To quantify their deviation from robust optimality, for each of these estimators, we determine the finite sample breakdown point, the influence function, as well as statistical accuracy measured by asymptotic bias, variance, and mean squared error - all evaluated uniformly on shrinking convex contamination neighborhoods.