= Local influence versus replacement breakdown
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>.<The Influence Curve and Its Role in Robust Estimation by Frank R. Hampel>[Frank R. Hampel, The Influence Curve and Its Role in Robust Estimation, §4, p. 386: exact asymptotic curve supplemented by behavior at other distributions, especially empirical ones]{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 $\varepsilon\uparrow1/2$ 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.
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