Uniqueness and Fréchet differentiability of functional solutions to maximum likelihood type equations. (English) Zbl 0541.62023

Author’s summary: 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 Fréchet differentiable and robust.
When several solutions exist, conditions on the loss criterion used to select the root for the statistic ensure Fréchet differentiability. An interesting example of a Fréchet 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.
Reviewer: D.Plachky


62F35 Robustness and adaptive procedures (parametric inference)
62E20 Asymptotic distribution theory in statistics
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