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An M-estimator based on an estimating equation chooses a consistent root ofIts population target $T(F)={\theta }_F$ solves ${\mathbb{E}}_F\psi (X,{\theta }_F)=0$. For a scalar parameter, suppose the root is locally unique andFor the contaminated distribution $F_{\epsilon }=(1-\epsilon )F+\epsilon {\delta }_x$, differentiate its population equation at zero contamination. The influence function satisfiesConsequentlyThe constant of proportionality is independent of $x$. A local expansion of the sample estimating equation similarly givesThe central limit theorem yields asymptotic normality with the sandwich variance of an M-estimatorThus the first-order variance of the unscaled estimator is $V/n$. For a vector parameter the same differentiation gives $\mathrm{IF}=A^{-1}\psi$ and $V=A^{-1}B{A}^{-\mathsf{T}}$, with $A=-\mathbb{E}{\partial }_{\theta }\psi$ and $B=\mathbb{E}\psi {\psi }^{\mathsf{T}}$.

For the location problem, take the target location to be zero and let $Z$ have the standard normal distribution, with density $\varphi$ and distribution function $\Phi$. An odd score has $\mathbb{E}\psi (Z)=0$. For a differentiable score, integration by parts givesThe last expression also gives the appropriate derivative of the population equation for bounded monotone scores with jumps, by differentiating the shifted normal density. Assume $A>0$ and put $h=\psi /A$. ThenMultiplying a score by a positive constant changes neither its estimator nor this normalized influence function.

First establish feasibility. The bound impliesso necessarily $C\ge \sqrt{\pi /2}$. For $C>\sqrt{\pi /2}$, let $K>0$ solveThis solution is unique: the ratio tends to $\sqrt{\pi /2}$ as $K\downarrow 0$, tends to infinity as $K\to \infty$, and has positive derivative becauseTake the Huber score ${\psi }_K(x)=\max (-K,\min (x,K))$. Its derivative is one on $(-K,K)$ and zero outside, so $A=p_K$. HenceIt is odd, nondecreasing, meets the bound, and satisfies $\mathbb{E}[Z{h}_K(Z)]=1$.

To prove optimality, put $a=1/p_K$. For each fixed $x$, the minimum of $u^2-2axu$ over $|u|\le C$ is attained by projecting $ax$ onto $[-C,C]$, which is exactly $h_K(x)$. Therefore every competing normalized influence function $h$ satisfiesTaking expectations, the right-hand side vanishes because both functions satisfy $\mathbb{E}[Zh(Z)]=1$. Thus $\mathbb{E}{h}^2\ge \mathbb{E}{h}_K^2$. This proves the optimal bounded influence function for normal location, even over the larger class of all bounded normalized functions, and hence over the required odd nondecreasing class. Its minimum asymptotic variance isThe numerator is $\mathbb{E}[Z^2{1}_{\{|Z|\le K\}}]+K^2\mathrm{Pr}(|Z|>K)$.

At the included boundary $C=\sqrt{\pi /2}$, equality in the feasibility bound forces $h(x)=C\mathrm{sgn}(x)$ almost everywhere. This is the influence function of the sample median, with $V=\pi /2$. It is obtained from the rescaled scores ${\psi }_K/K$ as $K\downarrow 0$. The boundary solution is the sign score, or equivalently the median estimator. Literal substitution of $K=0$ in the clipped score gives the identically zero function and no identifiable estimating equation; the printed clipped formula therefore requires this limiting interpretation at equality.

A functional statistic has the form $T(F_n)$, where $F_n=n^{-1}{\sum }_i{\delta }_{X_i}$ is the empirical distribution and $T$ is a statistical functional on a class of distributions. It separates the population quantity $T(F)$ from its sample estimate. Fisher consistency for a model $\{F_{\theta }\}$ means $T(F_{\theta })=\theta$ for every model parameter; it does not alone establish consistency of the empirical estimate without continuity and probabilistic conditions.

The influence function measures infinitesimal contamination at a point:when this derivative exists. A bounded influence function limits first-order sensitivity to an arbitrarily remote observation.

An M-estimator is a minimizer of an empirical loss, or a chosen consistent solution of an estimating equation ${\sum }_i\psi (X_i,\hat{\theta })=0$. Its population functional solves ${\mathbb{E}}_F\psi (X,T(F))=0$. Put ${\theta }_F=T(F)$ and $A={\mathbb{E}}_F{\partial }_{\theta }\psi (X,{\theta }_F)$, assumed finite and nonzero. Differentiating the contaminated population equation givessince the unperturbed expected score is zero. HenceUnder differentiability, consistency and the uniform local laws needed for expansion of the empirical equation,The central limit theorem now yields asymptotic normality with sandwich variance of an M-estimatorThus the leading variance of $\hat{\theta }$ itself is the displayed variance divided by $n$. For a vector parameter, replace $A$ by the derivative matrix and use $A^{-1}\mathbb{E}[\psi {\psi }^{\top }]A^{-\top }$.

In a location model with known scale $s$, choose $\psi (x,\theta )={\psi }_0((x-\theta )/s)$. For a symmetric reference distribution, an odd score is Fisher consistent. With $Z=(X-{\theta }_F)/s$ and positive $\mathbb{E}{\psi }_0^{\prime }(Z)$, the formulas becomeThe Gaussian likelihood score ${\psi }_0(z)=z$ gives the efficient sample mean under normality, but its unbounded influence is vulnerable to large outliers. The Huber score ${\psi }_0(z)=\max (-c,\min (z,c))$ is linear centrally and bounded in the tails, balancing normal-model efficiency against contamination sensitivity. Its denominator is $\mathbb{P}(|Z|<c)$ for a continuous law. The sign score gives the median and needs a nonsmooth derivation using the density at the median, rather than incorrectly setting its ordinary derivative to zero. Redescending scores can reject very extreme observations, but may have multiple roots and require a consistent root-selection rule.

Thus appropriate choices seek Fisher consistency, a nonzero stable derivative, finite variance, reasonable reference-model efficiency and bounded influence. Robust scale estimation may be needed for the tuning constant; joint location-scale estimation uses the matrix influence calculation rather than treating an estimated scale as known. Bounded influence describes local robustness and is not by itself a global breakdown point theorem.

At the standard normal distribution, symmetry and the odd Huber score give population root zero. For $0<b<\infty$, the derivative of the clipped score is $1_{\{|x|<b\}}$ except at its two corners, which have zero normal probability. Put $m_b=2\Phi (b)-1>0$ and write $\varphi$ for the standard normal density. The sandwich variance of an M-estimator has numeratorUsing ${\varphi }^{\prime }(z)=-z\varphi (z)$ and integration by parts,ThereforeAs checks, it tends to $1$ as $b\to \infty$, the sample mean limit, and to $\pi /2$ as $b\downarrow 0$, the sample median limiting variance. The estimator variance for a sample of size $n$ is approximately this quantity divided by $n$.

A scalar M-estimator is a selected root of $n^{-1}{\sum }_i\psi (Y_i,\hat{\theta })=0$, or equivalently a minimizer of a differentiable empirical objective whose derivative is this estimating equation. Its population statistical functional $T(F)={\theta }_F$ solves $\int \psi (x,{\theta }_F)dF(x)=0$. Assume local uniqueness, consistency of the sample root, a valid differentiable local expansion, finite ${\mathbb{E}}_F{\psi }^2$, andThese conditions include domination sufficient to differentiate under the integral and the required local uniform control of the sample derivative.

For $F_{\epsilon }=(1-\epsilon )F+\epsilon {\delta }_z$, differentiate the population estimating equation at $\epsilon =0$. Since its uncontaminated integral is zero,Thus the influence function is $\mathrm{IF}(z;T,F)=-\psi (z,{\theta }_F)/A$, with a proportionality constant independent of $z$.

Expanding the sample equation at its population root givesThe summands have mean zero. The central limit theorem and Slutsky's theorem therefore yield the scalar sandwich variance of an M-estimatorThis $V$ is the limiting variance of the $\sqrt{n}$-scaled error; the estimator's variance is approximately $V/n$. In a correctly centred location family with $\psi (x,\theta )=\psi (x-\theta )$, let $m={\mathbb{E}}_F{\psi }^{\prime }(X-\theta )$. Then $A=-m$, so the influence function is $\psi (z-\theta )/m$ and $V=\mathbb{E}{\psi }^2/m^2$. The minus sign from differentiating $x-\theta$ is essential.

The clipped Huber score in the subsequent subparts requires $0<b<\infty$. The PDF only displays an upper bound: at $b=0$ the estimating function vanishes identically, and at $b<0$ it is a constant with no root. A positive clipping threshold is the regular interpretation.