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Statistical expansions and locally uniform Fréchet differentiability

Part of: Distribution theory Nonparametric inference

Published online by Cambridge University Press:  09 April 2009

T. Bednarski ,
B. R. Clarke  and
W. Kolkiewicz
T. Bednarski
Affiliation:
Institute of MathematicsPolish Academy of Sciences Warsaw, Poland
B. R. Clarke
Affiliation:
Institute of MathematicsPolish Academy of Sciences Warsaw, Poland
W. Kolkiewicz
Affiliation:
Murdoch UniversityMurdoch, WA 6150, Australia
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Abstract

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Estimators which have locally uniform expansions are shown in this paper to be asymptotically equivalent to M-estimators. The M-functionals corresponding to these M-estimators are seen to be locally uniformly Fréchet differentiable. Other conditions for M-functionals to be locally uniformly Fréchet differentiable are given. An example of a commonly used estimator which is robust against outliers is given to illustrate that the locally uniform expansion need not be valid.

Keywords

infinitesimal neighbourhoodsFréchet differentiabilitystrong expansionsM-functionals

MSC classification

Secondary: 62E20: Asymptotic distribution theory 62G35: Robustness
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 1 , February 1991 , pp. 88 - 97
Copyright
Copyright © Australian Mathematical Society 1991

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