= Influence function of a Huber location estimator
{title2=$\operatorname{IF}(x;T,F)=s\psi_k((x-\theta)/s)/A$}
Fix a scale $s>0$ and cutoff $k>0$. Define the population <Huber location estimator> by the root
$$
\mathbb E_F\psi_k\!\left(\frac{X-\theta}{s}\right)=0,\qquad \theta=T(F),\qquad
A=\mathbb E_F\psi_k'\!\left(\frac{X-\theta}{s}\right)>0.
$$
Here $\psi_k$ 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 yields
$$
0=-\frac A s\operatorname{IF}(x;T,F)+\psi_k\!\left(\frac{x-\theta}{s}\right),\qquad
\operatorname{IF}(x;T,F)=\frac{s\psi_k((x-\theta)/s)}{A}.
$$
The 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>.<Robust Estimation of a Location Parameter by Peter J. Huber>[Peter J. Huber, Robust Estimation of a Location Parameter, §3, p. 78: derivative normalization; §4(iii), p. 79: clipped location criterion and central-mass denominator]{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:
$$
\operatorname{IF}(x;T,F)=\frac{s\psi_k(r_x)-B\operatorname{IF}(x;S,F)}{A},\qquad
r_x=\frac{x-\theta}{s},\quad B=\mathbb E_F\left[r_X\psi_k'(r_X)\right].
$$
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 (statistics)> 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.<Robust Estimation of a Location Parameter by Peter J. Huber>[Peter J. Huber, Robust Estimation of a Location Parameter, §11, pp. 95-96: estimating location when normal-component scale and contamination fraction are unknown]{ref}
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