The Annals of Statistics

A Note on Differentials and the CLT and LIL for Statistical Functions, with Application to $M$-Estimates

Dennis D. Boos and R. J. Serfling

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Abstract

A parameter expressed as a functional $T(F)$ of a distribution function (df) $F$ may be estimated by the "statistical function" $T(F_n)$ based on the sample df $F_n$. For analysis of the estimation error $T(F_n) - T(F)$, we adapt the differential approach of von Mises (1947) to exploit stochastic properties of the Kolmogorov-Smirnov distance $\sup_x|F_n(x) - F(x)|$. This leads directly to the central limit theorem (CLT) and law of the iterated logarithm (LIL) for $T(F_n) - T(F)$. The adaptation also incorporates innovations designed to broaden the scope of statistical application of the concept of differential. Application to a wide class of robust-type $M$-estimates is carried out.

Article information

Source
Ann. Statist., Volume 8, Number 3 (1980), 618-624.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345012

Digital Object Identifier
doi:10.1214/aos/1176345012

Mathematical Reviews number (MathSciNet)
MR568724

Zentralblatt MATH identifier
0434.62022

JSTOR
links.jstor.org

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 62G35: Robustness

Keywords
Differentials functionals statistical functions asymptotic normality law of the iterated logarithm $M$-estimates

Citation

Boos, Dennis D.; Serfling, R. J. A Note on Differentials and the CLT and LIL for Statistical Functions, with Application to $M$-Estimates. Ann. Statist. 8 (1980), no. 3, 618--624. doi:10.1214/aos/1176345012. https://projecteuclid.org/euclid.aos/1176345012


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