 M-estimator

An M-estimator chooses a parameter by minimizing an empirical objective: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 ${\sum }_i\psi (x_i,\hat{\theta })=0$, 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 $\theta =\pm 1$. 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.^[ref]