The Annals of Statistics

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

R. M. Dudley

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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.

Article information

Ann. Statist., Volume 22, Number 1 (1994), 1-20.

First available in Project Euclid: 11 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G30: Order statistics; empirical distribution functions
Secondary: 58C20: Differentiation theory (Gateaux, Fréchet, etc.) [See also 26Exx, 46G05] 26A45: Functions of bounded variation, generalizations 60F17: Functional limit theorems; invariance principles

Frechet derivative compact derivative Hadamard derivative Gateaux derivative Bahadur-Kiefer theorems Orlicz variation


Dudley, R. M. The Order of the Remainder in Derivatives of Composition and Inverse Operators for $p$-Variation Norms. Ann. Statist. 22 (1994), no. 1, 1--20. doi:10.1214/aos/1176325354.

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