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Uniqueness and Fréchet differentiability of functional solutions to maximum likelihood type equations

Clarke, B.R. (1983) Uniqueness and Fréchet differentiability of functional solutions to maximum likelihood type equations. Annals of Statistics, 11 (4). pp. 1196-1205.

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    Abstract

    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.

    Publication Type: Journal Article
    Publisher: Institute of Mathematical Statistics
    Copyright: © Institute of Mathematical Statistics
    URI: http://researchrepository.murdoch.edu.au/id/eprint/4699
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