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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.

WriteIf $X_{+}\sim F_{+}$, then $X_{+}-2b_1\sim F_{-}$ because $F_{-}(t)=F_{+}(t+2b_1)$. The translation-invariant estimator property givesFor every real $u$,Taking $u={\mu }_{+}$ shows that every admissible estimator has maximum asymptotic bias at least $b_1$ on the pair $\{F_{+},F_{-}\}$. ThereforeTogether with part a, this proves the minimax asymptotic bias optimality of the median.

Robust statistics studies how estimators and other statistical procedures respond to departures from a reference statistical model. Such departures can include a small fraction of outliers, a heavier-tailed probability distribution, or an incorrect assumed shape. A robustness statement specifies both the permitted departure and the quantity whose change is controlled.

For data $(0,1,2,3,M)$ with $M\ge 4$, the sample median remains $2$, whereas the sample mean is $(6+M)/5$ and grows without bound. This does not make the sample median an estimator of every desired expected value: the mean and median of an asymmetric probability distribution can differ. The estimand must therefore be specified before judging robustness.

Influence functions describe infinitesimal changes, breakdown points describe large finite-sample failures, and minimax asymptotic bias compares worst cases over a neighborhood. M-estimators connect these questions with an explicit loss function. Peter J. Huber's 1964 location paper developed a systematic treatment of estimation under contamination.^[ref]

For a location estimator, the property historically called translation invariance in this setting is more explicitly called translation equivariance:Changing the origin of the measurements changes the estimated location by exactly the same amount. Both the sample mean and sample median satisfy this identity. For example, adding $10$ to every observation adds $10$ to either estimated location.

Strict invariance would instead leave the numerical answer unchanged. The sample variance has that different property because shifting every observation also shifts their sample mean, leaving all centered differences unchanged. The distinction matters when specifying the admissible estimators in a minimax asymptotic bias problem. The older translation-invariant terminology appears in Peter J. Huber's original location analysis.^[ref]