= 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 $\mathcal N_\varepsilon(F)$ and maximum asymptotic displacement
$$
B_\varepsilon(T,F)=\sup_{G\in\mathcal N_\varepsilon(F)}|T(G)-T(F)|.
$$
<Minimax 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 $(1-\varepsilon)\mu+\varepsilon M$ when the contaminating <probability distribution> is a <Dirac measure> at $M$. Its maximum displacement is therefore infinite for every $\varepsilon>0$. A <median> can have finite maximum displacement for $\varepsilon<1/2$. 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>.
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