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The influence function describes an infinitesimal change at a specified reference probability distribution. The replacement breakdown point asks how many observations in a particular finite sample can be replaced before an estimator becomes unbounded. The finite-sample maximum bias records the intervening changes. These are different questions: a bounded influence function at one baseline is not a general bound over all epsilon-contamination neighborhoods, nor a proof of a positive replacement breakdown point.^[ref]

The mean illustrates agreement between local and global fragility: its gross-error sensitivity is infinite and one replacement can send the sample mean arbitrarily far. The median has finite gross-error sensitivity under the stated positive-density assumptions, while its sample median withstands fewer than about half the observations being replaced under ordinary tie conventions. Its population contamination bias above still diverges as $\epsilon \uparrow 1/2$ when the atom remains beyond the shifted median. Here $x$ must also move outward; taking that limit with a fixed finite atom would invalidate the displayed quantile formula.

For scalar location with a fixed positive scale, the Huber score has a direct global argument: if strictly more than half of the sample stays fixed, then beyond every retained observation by more than $ks$, their saturated scores outweigh all possible replaced scores. Every root therefore stays in a bounded interval. This is a separate finite-replacement argument; estimating scale changes its premises. A smooth bounded influence function is valuable local protection, but global robustness also depends on the full estimating rule, scale choice, and root selection.

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.