= M-estimator
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An <M-estimator> chooses a parameter by minimizing an empirical objective:
$$
\widehat\theta\in\operatorname*{argmin}_{\theta}\sum_{i=1}^n\rho(x_i,\theta).
$$
Here $\rho$ is a <loss function>. For scalar location, <squared-error loss> gives the <sample mean>, <absolute-error loss> gives a <sample median>, and <Huber loss> gives a <Huber location estimator>.
At an interior <minimum> of a <differentiable> objective, the <estimating equation> is $\sum_i\psi(x_i,\widehat\theta)=0$, where $\psi=\partial_\theta\rho$. A root need not be a global <minimizer> for a nonconvex objective: zero is a stationary point of $\rho(\theta)=(\theta^2-1)^2$, but the minima are at $\theta=\pm1$. For a <differentiable> <convex function>, a zero <derivative> is sufficient for global minimality.
Thus the loss-minimization and estimating-equation formulations require an explicit relation between their solutions. The <sandwich variance of an M-estimator> describes regular estimating-equation roots; choosing a robust loss does not itself establish the regularity or uniqueness needed for that calculation. <Peter J. Huber>'s original location criterion is a central historical example.<Robust Estimation of a Location Parameter by Peter J. Huber>[Section 4(iii), clipped location criterion]{ref}
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