 Minimax asymptotic bias

Suppose a statistical functional $T(G)$ is the limiting answer of an estimator when observations have probability distribution $G$. Around a reference probability distribution $F$, define a neighborhood ${\mathcal{N}}_{\epsilon }(F)$ and maximum asymptotic displacementMinimax asymptotic bias minimizes this quantity over a specified admissible class of estimators. Calibration on the reference family and restrictions such as translation equivariance prevent a constant answer from being an artificial solution.

For an epsilon-contamination neighborhood, the mean changes from $\mu$ to $(1-\epsilon )\mu +\epsilon M$ when the contaminating probability distribution is a Dirac measure at $M$. Its maximum displacement is therefore infinite for every $\epsilon >0$. A median can have finite maximum displacement for $\epsilon <1/2$. This compares a finite contamination fraction; it is distinct from bounding an influence function, which concerns the derivative at zero contamination. The neighborhood and admissible class are part of the problem, rather than a universal choice of best estimator.