Solutions of simultaneous equations of the maximum likelihood type or $M$-estimators can be represented as functionals. Existence and uniqueness of a root in a local region of the parameter space are proved under conditions that are easy to check. If only one root of the equation exists, the resulting statistical functional is Frechet differentiable and robust. When several solutions exists, conditions on the loss criterion used to select the root for the statistic ensure Frechet differentiability. An interesting example of a Frechet differentiable functional is the solution of the maximum likelihood equations for location and scale parameters in a Cauchy distribution. The estimator is robust and asymptotically efficient.
"Uniqueness and Frechet Differentiability of Functional Solutions to Maximum Likelihood Type Equations." Ann. Statist. 11 (4) 1196 - 1205, December, 1983. https://doi.org/10.1214/aos/1176346332