= Huber location estimator
{c}
{title2=$\widehat\theta_{k,s}$}
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 $x_1,\ldots,x_n$, a <Huber location estimator> is a <minimizer> of the summed <Huber loss>, equivalently a root of its <estimating equation>:
$$
Q_{k,s}(\theta)=\sum_{i=1}^n\rho_k\!\left(\frac{x_i-\theta}{s}\right),\qquad
\widehat\theta_{k,s}\in\operatorname*{argmin}_{\theta\in\mathbb R}Q_{k,s}(\theta),\qquad
\sum_{i=1}^n\psi_k\!\left(\frac{x_i-\widehat\theta_{k,s}}s\right)=0.
$$
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.<Robust Estimation of a Location Parameter by Peter J. Huber>[§4(iii), printed p. 79: quadratic-central, linear-tail criterion and clipping around the fitted location]{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> are
$$
(x_i-2.5)_{i=1}^6=(-2.5,-1.5,-0.5,0.5,1.5,37.5),\qquad
(\psi_2(x_i-2.5))_{i=1}^6=(-2,-1.5,-0.5,0.5,1.5,2).
$$
Their 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 $50/6\simeq8.3333$, 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 $n\geq1$, $Q_{k,s}(\theta)$ 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.
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