= Robust estimator
{title2=$\hat\theta$}
A <robust estimator> is an <estimator> whose behavior remains acceptably stable under specified departures from an assumed <statistical model>. “Stable” must identify a criterion, such as limited bias under <contamination (statistics)>, bounded response to a single extreme observation, or resistance to replacing a fraction of the sample. It does not mean that the <estimator> is accurate under every possible <probability distribution>.
For example, consider data $(-1,0,1,M)$ with $M\geq1$. The <sample mean> is $M/4$, whereas the <sample median>, using the midpoint convention for an even-sized sample, is $1/2$. Sending $M$ to infinity makes the <sample mean> diverge without changing that <sample median>. The example concerns location estimation: the population <median> and population <mean> need not coincide outside a symmetric reference model.
One formal model is <Huber contamination>:
$$
F_\varepsilon=(1-\varepsilon)F+\varepsilon G,
$$
where $F$ is a reference <probability distribution>, $G$ is an unspecified contaminating <probability distribution>, and $\varepsilon$ is its proportion. A robustness claim can bound an <estimator>’s worst bias over allowed $G$, rather than only calculating its <variance> when the reference model is exactly true. <Peter J. Huber>’s original location paper treats this tradeoff systematically.<Robust Estimation of a Location Parameter by Peter J. Huber>[Introduction, contaminated normal location models; section 4(iii), clipped location criterion]{ref}
A <Huber location estimator> illustrates how stability enters the calculation. With a specified positive scale $s$, solve
$$
\sum_{i=1}^n\psi_c\!\left(\frac{x_i-\widehat\theta}{s}\right)=0,
\qquad \psi_c(u)=\max(-c,\min(c,u)),\quad c>0.
$$
Large residuals have bounded scores, so one increasingly extreme point cannot contribute an arbitrarily large term to this equation. By contrast, the squared-error score grows linearly with the residual and yields the <sample mean>. A practical analysis must also state how $s$ is obtained: a scale estimate that itself explodes under <outliers> can undermine the intended protection.
Different criteria answer different questions. An <influence function> studies an infinitesimal perturbation of a distribution, while a <breakdown point> concerns a finite amount of sample corruption. A <robust estimator> can sacrifice some model-specific <statistical efficiency> to improve performance away from that model, but the amount depends on the chosen tuning and departure. Reporting the target parameter, reference model and robustness criterion makes the claim testable.
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