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Consider the population statistical model $F_{\epsilon ,x}=(1-\epsilon )N(0,1)+\epsilon {\delta }_x$. The following are deterministic statistical functional calculations, not observed contamination experiments. Compare the actual location bias with the linear approximation $\epsilon \mathrm{IF}(x;T,F)$. The mean bias is exactly $\epsilon x$. For positive $x$ exceeding the shifted median and $0\le \epsilon <1/2$, the generalized-inverse median bias is exactlyThis formula applies to both $x=1$ and $x=8$ throughout the plotted interval $0\le \epsilon \le 0.1$. If the contaminating atom lies inside the quantile jump containing probability $1/2$, the generalized-inverse median instead equals that atom.

For the Huber location estimator with $s=1$ and $k=1.345$, solve its actual population equation, rather than substituting the influence function:Writing $a=t-k$ and $b=t+k$, integration of the normal distribution giveswhere $\varphi$ is the probability density function of $N(0,1)$. Its derivative is $-[\Phi (t+k)-\Phi (t-k)]<0$, so the mixture score is strictly decreasing for $\epsilon <1$. For $\epsilon <1/2$, its limits at negative and positive infinity have opposite signs even with arbitrary distant point contamination. The displayed examples use the unique root, evaluated by the bisection method to numerical precision.

At $x=8$ and $\epsilon =0.01$, the exact mean, median, and Huber location estimator biases are $0.080000$, $0.012660$, and $0.016541$; the influence function predictions are $0.080000$, $0.012533$, and $0.016375$. At $\epsilon =0.05$, they become $0.400000$, $0.066012$, and $0.086241$, versus predictions $0.400000$, $0.062666$, and $0.081875$. The bounded alternatives control the effect of the distant atom, but their actual response curves bend away from the tangent.

The nearby atom $x=1$ gives a useful comparison: at $\epsilon =0.05$, the Huber location estimator bias is $0.060237$, slightly below its linear approximation $0.060874$, whereas the median bias is still $0.066012$. The direction and size of approximation error depend on the contaminating location. Even accurate local influence calculations must specify $F$, the scale, the cutoff, and the actual contamination fraction.

The Huber location estimator solves ${\sum }_i{\psi }_k(x_i-\theta )=0$, where ${\psi }_k$ clips its argument to $[-k,k]$.

More generally, fix a known scale $s>0$ and a dimensionless threshold $k>0$ before fitting. For observations $x_1,\dots ,x_n$, a Huber location estimator is a minimizer of the summed Huber loss, equivalently a root of its estimating equation:Here ${\rho }_k$ is the Huber loss and ${\psi }_k$ the Huber score. The physical clipping distance is $ks$, in the observations' units; the initial formula uses $s=1$. A large observation keeps contributing a bounded signed score rather than being discarded. The original clipped location criterion appears in Peter J. Huber's 1964 paper.^[ref]

For the original data $(0,1,2,3,4,40)$, take $s=1$ and $k=2$. At $\theta =2.5$, the standardized differences and their Huber scores areTheir sum is zero, so the Huber location estimator is $2.5$. Four scores remain in the central linear region, making the root unique. By comparison, the sample mean is $50/6\simeq 8.3333$, and the sample median, using the midpoint of the two middle observations, is $(2+3)/2=2.5$. Agreement with the sample median is a feature of these data, rather than an identity between the two estimators.

The geometry explains both existence and possible nonuniqueness. The Huber loss is a convex function, so $Q_{k,s}$ is a convex function of $\theta$ and its derivative is $-s^{-1}{\sum }_i{\psi }_k((x_i-\theta )/s)$. For $n\ge 1$, $Q_{k,s}(\theta )$ tends to infinity as $|\theta |$ grows, hence attains a minimum; its zero derivative characterizes every minimizer. Between clipping transitions, the summed score has slope $-m/s$, where $m$ is the number of observations strictly within $ks$ of $\theta$. A root with $m>0$ is unique. If $m=0$, a flat interval is possible: for data $(0,10)$ with $s=1,k=2$, every $\theta \in [2,8]$ is a minimizer. A complete single-valued estimator then needs a selection rule, such as the interval's midpoint.

A bounded Huber score limits one observation's contribution to the estimating equation. The influence function of a Huber location estimator additionally divides by the population's central mass, so the size of its bound depends on the reference probability distribution. Clipping does not by itself make the bias of an estimator zero or prove a universal breakdown point; the separate local influence versus replacement breakdown argument specifies what fixed scale permits. Estimating $s$ from the same data changes the procedure and requires separate analysis.

Hold $(0,1,2,3,4)$ fixed and move only the sixth observation $M$. For $M\ge 5$, the sample mean is $(10+M)/6$, the sample median is $2.5$, and the Huber location estimator with fixed $s=1,k=2$ stays at $2.5$. Once $M-2.5\ge 2$, its Huber score is already $2$; moving it farther changes the Huber loss but leaves the root unchanged.

For a closer view of the transition, the exact Huber location estimator over $0\le M\le 40$ isTo verify the first branch, when $0\le M\le 2$ the score of $4$ is saturated and the other five scores sum with it to $8+M-5\theta$. When $2\le M\le 4.5$, the score of $0$ is saturated instead, giving the same expression. At $M=4.5$, the moving observation saturates too, and the equation becomes $10-4\theta =0$. The piecewise formula is continuous at the transition.

The rising sample mean comes from an unbounded linear score, the flat distant-outlier response from a bounded Huber score. This is a finite-data response curve with a specified scale and threshold, rather than a population influence function or a general claim about arbitrary numbers of outliers.

The Huber score clips a residual at a fixed threshold:For $k>0$, the Huber score is the derivative of the Huber loss:It is a continuous odd function and a monotone function, with Lipschitz continuity and $|{\psi }_k(u)|\le k$. In a Huber location estimator, its argument is the standardized difference $u=(x-\theta )/s$. Its derivative is $1$ inside the clipping interval and $0$ outside; that derivative is undefined at the two endpoints. The score itself remains continuous there. This estimating score need not be the score function of a separately specified statistical model, nor is it already a normalized influence function.

An influence function asks how much a population answer moves when an infinitesimal fraction of its probability distribution is relocated to a specified point. It distinguishes the unbounded response of the mean from the bounded response of the median and the Huber location estimator. The normal-location influence curves and finite contamination bias of normal location below make that difference quantitative.

Let $T$ be a real-valued statistical functional, $F$ a reference probability measure, and ${\delta }_x$ the Dirac measure concentrated at $x$. Along the epsilon-contamination neighborhood path $F_{\epsilon ,x}=(1-\epsilon )F+\epsilon {\delta }_x$, define the right derivativeThe name influence curve refers to the same object. The derivative is in the signed direction ${\delta }_x-F$, so the calculation removes baseline mass as well as adding a Dirac measure. The definition is a limit for each fixed $x$; it does not automatically give a remainder uniform over arbitrarily distant $x$.^[ref]

The statistical functional must be defined on these contaminated probability measures, and the right derivative must exist and be finite. For a mean, a finite first absolute moment suffices. For a median, require a unique baseline median and a probability density function continuous and strictly positive there. For a smooth M-estimator, require an isolated population root, justified differentiation of its expected value, and a nonzero derivative with respect to the fitted parameter. These are specific conditions for the formulas below. Existence of the pointwise influence function alone proves neither an asymptotic linear representation nor pathwise differentiability of a statistical functional; each needs control of its own remainder and admissible statistical paths.

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]

An M-estimator minimizes an empirical objective ${\sum }_i\rho (x_i,\theta )$ or, when differentiable, solves an estimating equation ${\sum }_i\psi (x_i,\theta )=0$.

The root calculation for the Huber location estimator givesSince $|{\psi }_b(z)|\le b$ and equality is attained for $|z|\ge b$, its gross-error sensitivity isThus it is a B-robust estimator at the normal reference distribution. At a general centred distribution the same conclusion requires $m={\mathbb{E}}_F{\psi }_b^{\prime }(X)>0$, in which case the bound is $b/m$. Boundedness of the score alone does not rescue a zero sensitivity denominator.

The normal location score is $x-\theta$. Under a bound on gross-error sensitivity, the variance-minimizing influence curve clips this score. For $b>0$, define the Huber scoreThe optimal B-robust estimator is the Huber location estimator ${\hat{\theta }}_b$ defined byEquivalently, it has the explicit optimization formwhere the Huber loss isAt $N(\theta ,1)$ its normalized influence function is

The Huber location estimator usesaway from the two corners. At $F=N(\theta ,1)$, symmetry gives $T(F)=\theta$, and for $Z\sim N(0,1)$,Hencewhich is bounded in $x$.

The gross-error sensitivity is ${\gamma }^{\ast }(T,F)={\sup }_x|\mathrm{IF}(x;T,F)|$. At $N(\theta ,1)$, the influence function of the sample median has magnitude $1/(2\phi (0))=\sqrt{\pi /2}$, soFor the Huber location estimator, ${\psi }_k(u)=\max (-k,\min (u,k))$ and $\mathbb{E}{\psi }_k^{\prime }(Z)=\mathbb{P}(|Z|\le k)=2\Phi (k)-1$, givingFor the symmetric normal law, the trimmed population mean is $\theta$. Writing $q_{\gamma }={\Phi }^{-1}(1-\gamma )$ in the supplied influence function gives