What is it about?
This paper supplies conditions A which are used to show weak continuity and Frechet differentiability of functional solutions to maximum likelihood type equations, also known as M-estimators, when the defining psi function is continuously differentiable and bounded. It was thought by Huber (1981) "Robust Statistics" Wiley publishers, that..."Unfortunately the concept of Frechet differentiability appears to be too strong in too many cases, the Frechet derivative does not exist, and even if it does, the fact is difficult to establish". The current paper gives conditions which are easily satisfied for M-estimators with bounded and smooth psi - functions. Indeed M-estimators satisfying these conditions are robust.
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Why is it important?
The paper makes a break through in understanding of what defines a robust M-estimator. Conditions A have since been used to provide robustness in testing and in general inference. This study supports the intuitive concepts of robustness that were alluded to but not specifically defined in the PhD thesis of Hampel (1968), Berkeley Statistics Department.
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This page is a summary of: Uniqueness and Frechet Differentiability of Functional Solutions to Maximum Likelihood Type Equations, The Annals of Statistics, December 1983, Institute of Mathematical Statistics,
DOI: 10.1214/aos/1176346332.
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