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Nonsmooth analysis and Fréchet differentiability of M-functionals. (English) Zbl 0581.60005
Existence of the strict Fréchet derivative for statistical functionals is an advantage for it implies amongst other things properties of asymptotic normality and regular behaviour of statistical functionals in infinitesimal neighbourhoods, indicative of robustness. The derivative is stronger than weaker forms of derivative, proposed by other authors, which mostly rely on probabilistic error terms.
In this paper techniques of nonsmooth analysis are introduced to the theory of statistical expansions in order to establish conditions for existence of Fréchet derivatives for common choices of robust M- functionals where the defining \(\Psi\)-functions have ”sharp corners”. A new Glivenko-Cantelli result is also proved which is used to show one of the conditions to hold, and this may have wider applications.

MSC:
60B05 Probability measures on topological spaces
62E20 Asymptotic distribution theory in statistics
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