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March, 1994 The Order of the Remainder in Derivatives of Composition and Inverse Operators for $p$-Variation Norms
R. M. Dudley
Ann. Statist. 22(1): 1-20 (March, 1994). DOI: 10.1214/aos/1176325354

Abstract

Many statisticians have adopted compact differentiability since Reeds showed in 1976 that it holds (while Frechet differentiability fails) in the supremum (sup) norm on the real line for the inverse operator and for the composition operator $(F,G) \mapsto F \circ G$ with respect to $F$. However, these operators are Frechet differentiable with respect to $p$-variation norms, which for $p > 2$ share the good probabilistic properties of the sup norm, uniformly over all distributions on the line. The remainders in these differentiations are of order $\| \cdot \|^\gamma$ for $\gamma > 1$. In a range of cases $p$-variation norms give the largest possible values of $\gamma$ on spaces containing empirical distribution functions, for both the inverse and composition operators. Compact differentiability in the sup norm cannot provide such remainder bounds since, over some compact sets, differentiability holds arbitrarily slowly.

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R. M. Dudley. "The Order of the Remainder in Derivatives of Composition and Inverse Operators for $p$-Variation Norms." Ann. Statist. 22 (1) 1 - 20, March, 1994. https://doi.org/10.1214/aos/1176325354

Information

Published: March, 1994
First available in Project Euclid: 11 April 2007

zbMATH: 0816.62039
MathSciNet: MR1272072
Digital Object Identifier: 10.1214/aos/1176325354

Subjects:
Primary: 62G30
Secondary: 26A45 , 58C20 , 60F17

Keywords: Bahadur-Kiefer theorems , compact derivative , Frechet derivative , Gateaux derivative , Hadamard derivative , Orlicz variation

Rights: Copyright © 1994 Institute of Mathematical Statistics

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Vol.22 • No. 1 • March, 1994
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