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The Order of the Remainder in Derivatives of Composition and Inverse Operators for p-Variation Norms

  • R. M. Dudley
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

The theory of differentiable statisticals began with work of von Mises [e.g., von Mises (1936, 1947) and Filippova (1961)]. A nonlinear funtional T is defined, for examble, on distribution functions.Von Mises differentiated T at a distribution funtion F along lines. For T to have a (Gâteaux) derivative at F means that, in the direction of a function h,
$$T(F + th) = T(f) + tT^\prime(F)(h) + 0 (|t|) \quad {\rm as} \; t\rightarrow 0.$$
(1.1)
Here \(T^\prime(F)(.)\) is a bounded linear operator on funtions h, for example, of the form
$$T^\prime(F) (h) = \int g \quad dh \hbox{for some function} g (\hbox{depending on} F).$$
(1.2)

Key words and phrases

Fréchet derivative compact derivativ Hadamard derivative Gâteaux derivative Bahadur–Kiefer theorems Orlicz variation 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • R. M. Dudley
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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