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Fix a scale $s>0$ and cutoff $k>0$. Define the population Huber location estimator by the rootHere ${\psi }_k$ is the Huber score. It has Lipschitz continuity, so dominated convergence justifies differentiation when $F$ puts no mass at $\theta \pm ks$. Then $A={\mathbb{P}}_F(|X-\theta |<ks)$, and the nonzero derivative identifies the local population root. The implicit function theorem applies away from the contaminating score's kinks; its Lipschitz continuity also gives the first-order expansion at those kinks. Differentiating its contaminated equation yieldsThe normalization matters: the Huber score alone is not the influence function. The gross-error sensitivity is $sk/A$, while the rejection point of an influence function is infinite because the clipped tails stay nonzero. The original clipped estimating criterion and its derivative normalization also appear in the population analysis of M-estimators.^[ref]

Scale sets what counts as a large residual. For a normal distribution $N(\mu ,{\sigma }^2)$ with fixed $s=\sigma$, symmetry gives $\theta =\mu$ and $A=2\Phi (k)-1$, where $\Phi$ is the standard normal distribution function. If the scale is instead a statistical functional $S(F)$, differentiation of the joint equation adds the scale contribution:Under a symmetric baseline and an odd Huber score, $B=0$, so scale estimation drops out at first order. This cancellation needs regularity of $S$ and does not protect finite contamination changes: an inflated standard deviation enlarges the physical cutoff $ks$. A robust choice, such as a suitable Scale M-estimator, needs its own influence function and breakdown point checked. Fixing a numeric cutoff in raw units also sacrifices scale equivariance when units change.^[ref]

Population influence functions describe infinitesimal contamination; finite-sample analogues examine actual changes to an empirical sample. For its empirical distribution $F_n$, add an observation $z$ and define the sensitivity curveUnder suitable differentiability it approximates the influence function. A replacement version is ${\mathrm{RC}}_{n,i}(z)=n[T_n(y_1,\dots ,z,\dots ,y_n)-T_n(y)]$, approximately $\mathrm{IF}(z;T,F_n)-\mathrm{IF}(y_i;T,F_n)$. Their supremum magnitudes quantify finite one-observation sensitivity.

The empirical influence function instead substitutes $F_n$ directly in $\mathrm{IF}(z;T,F)$. For a differentiable M-estimator this gives $-\psi (z,T(F_n))/[n^{-1}{\sum }_i{\partial }_{\theta }\psi (y_i,T(F_n))]$. For quantile functionals, a discrete unsmoothed empirical law may not admit the required density derivative; a sensitivity curve or a smoothed empirical law is then preferable.

Global contamination is measured by finite-sample maximum bias $b_n(m;y)={\sup }_{z:d_H(z,y)\le m}|T_n(z)-T_n(y)|$, where $d_H$ counts replacements. Under the convention of the first fraction that causes unbounded bias,This is the finite-sample replacement breakdown point. The largest replacement fraction still guaranteed bounded is one grid step smaller, so the conventions must be specified. One replacement sends a sample mean arbitrarily far, giving $1/n$, while the ordinary scalar sample median has first-breaking fraction $\lceil n/2\rceil /n$ with the usual midpoint convention for even samples. Its limiting value is $1/2$. Local influence and global breakdown answer different questions; a bounded influence function is not by itself a proof of a high breakdown point.

Treatment contamination occurs when participants in one arm of a randomized controlled trial receive an intervention assigned to another arm. This is distinct from contamination in robust estimation. The intention-to-treat analysis still compares randomized assignments, but intervention receipt becomes less separated between the arms, often diluting the assignment effect. Receipt-based comparisons can introduce confounding because receipt need not be randomized.