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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.

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

By Peter J. Huber.

Publisher: Institute of Mathematical Statistics.

Published in The Annals of Mathematical Statistics, volume 35, number 1, pp. 73-101.

Online text: doi.org/10.1214/aoms/1177703732

Cited passages: Peter J. Huber, Robust Estimation of a Location Parameter, §11, pp. 95-96: estimating location when normal-component scale and contamination fraction are unknown; 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.

Cited passages: original journal wrapper and byline, March 1964 publication; §1, printed pp. 73–75; §2, printed p. 76, minimizing set; §4(iii), printed p. 79, piecewise criterion and clipping about the fitted location. The manuscript receipt date, 4 June 1963, precedes publication.

Inspected mirrored scan transcription: www.scribd.com/document/861209215/Huber-RobustEstimationLocation-1964

Robust statistics studies procedures whose behavior remains controlled under outliers and deviations from an assumed model.

A location estimator $T_n$ is translation-invariant whenfor every real shift $a$.