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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]

Write $y_i=|x_i|$. If $n$ is odd, the sample median $t=T_n$ equals one observation, with equally many $y_i$ below and above it; the corresponding terms $\mathrm{sign}(y_i/t-1)$ cancel and the median term is zero. If $n$ is even, the conventional median lies strictly between the two middle observations when the $y_i$ are distinct, so exactly half the terms are $-1$ and half are $+1$. In either caseHence $T_n$ is a Scale M-estimator.

A Scale M-estimator estimates the spread of centered data by solvingfor a chosen function $\psi$. If the center is estimated, residuals $x_i-\hat{\mu }$ replace $x_i$. The function and its normalization specify the population scale being estimated; an arbitrary choice need not have a positive or unique root.

For known center zero and at least one nonzero observation, choosing $\psi (u)=u^2-1$ gives $t=\sqrt{n^{-1}{\sum }_i{x}_i^2}$, the root mean square. This example is highly sensitive to an outlier. Bounded choices of $\psi$ can instead limit an observation's contribution to the estimating equation, but their influence function and breakdown point still require analysis.

If $\psi$ is an even function and the positive root is unique, multiplying the data by $c\ne 0$ multiplies the root by $|c|$: substitute $t_c=|c|t$ into the equation. This is scale equivariance, the expected response to changing measurement units.