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Here $d_H$ counts the observations replaced. This curve measures the largest change after at most $m$ arbitrary replacements. Under the first-unbounded convention, the replacement breakdown point is $n^{-1}\min \{m:b_n(m;y)=\infty \}$. The largest fraction still guaranteed bounded is one grid step smaller; those conventions must not be confused. A sample mean breaks after one replacement, while a scalar sample median with a fixed ordinary tie convention needs about half the sample replaced.

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.

Let $m=\lfloor \alpha n\rfloor$. If at most $m$ observations are replaced arbitrarily, every order statistic retained between ranks $m+1$ and $n-m$ remains between the minimum and maximum of the unreplaced observations. The trimmed mean is therefore bounded as the replacement values diverge.

If more than $m$ observations are replaced by a common value tending to $+\infty$, at least one replacement remains after the largest $m$ observations are trimmed, and the trimmed mean tends to $+\infty$. The analogous construction tends to $-\infty$. Thus the largest fraction of arbitrary replacements for which boundedness is guaranteed isThis is the convention for the finite-sample replacement breakdown point used in the question; the alternative convention based on the smallest breaking fraction reports $(m+1)/n$.

Suppose for contradiction that $\Vert \hat{\theta }(Z_{\gamma ,\tau }){\Vert }_2\le M$ uniformly in $\gamma ,\tau$. Choose $\gamma >M+1$, so part d applies to every fitted value and gives $L(\hat{\theta })\ge \rho (\tau )$. For this fixed $\gamma$, part c gives the competing boundwhich is independent of $\tau$. Since $\rho (\tau )\to \infty$, choose $\tau$ so that the lower bound exceeds this upper bound, contradicting optimality. Thus $n-h+1$ replacements can make the estimate unbounded, andTogether with part b, the replacement breakdown point is exactly $(n-h)/n$.

Let replacement tolerance mean the greatest number of observations that can be replaced while the estimator remains bounded. For $n=2m$, a sample with $m$ zeros and $m$ copies of $a$ is within $m$ replacements of both the all-zero sample and its translate by $a$. If an equivariant estimator tolerated $m$ replacements, it would remain within bounded distance of both $T(0^n)$ and $T(0^n)+a$, which is impossible as $a\to \infty$. Thus at most $m-1$ replacements are tolerable.

For $n=2m+1$, a sample with $m$ zeros and $m+1$ copies of $a$ is obtained from the all-zero sample by $m+1$ replacements and from the all-$a$ sample by $m$ replacements. Translation equivariance again forces breakdown by $m+1$ replacements. In both cases the finite-sample replacement breakdown point is at most