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Statistical expansions and locally uniform Fréchet differentiability

Part of: Nonparametric inference Distribution theory

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

References

Bednarski, T. (1985), ‘On minimum bias and variance estimation for parametric models with shrinking contamination’, Probab. Math. Statist. 6, 121–129.Google Scholar
Beran, R. (1977), ‘Minimum Hellinger distance estimates for parametric models’, Ann. Statist. 5, 445–463.Google Scholar
Bickel, P. J. (1981), ‘Quelques aspects de la statistique robuste’, Lecture Notes in Mathematics pp. 1–72, (Ecole d'Ete de Probabilités de Saint-Fleur IX, 1971).Google Scholar
Clarke, B. R. (1983), ‘Uniqueness and Fréchet differentiability of functional solutions to maximum likelihood type equations’, Ann. Statist. 11, 1196–1205.Google Scholar
Clarke, B. R. (1986), ‘Nonsmooth analysis and Fréchet differentiability of M-functionals, Probab. Th. Rel. Fields 73, 197–209.CrossRefGoogle Scholar
Cramér, H. and Wold, H. (1936), ‘Some theorems on distribution functions’, J. London Math. Soc. 11, 290–295.CrossRefGoogle Scholar
Fernholz, L. T. (1983), Von Mises calculus for statistical functionals, (Lecture Notes in Statistics 19, Springer Berlin, Heidelberg, New York).Google Scholar
Gill, R. D. (1989), ‘Non- and semi-parametric maximum likelihood estimators and the von Mises method (Part 1)’, Scand. J. Statist. 16, 97–128.Google Scholar
Hampel, F. R. (1974), ‘The influence curve and its role in robust estimation’, J. Amer. Statist. Assoc. 62, 1179–1186.Google Scholar
Huber-Carol, C. (1970), Etude asymptotique de tests robustes, (These a l'Ecole Polytechnique Federale de Zürich).Google Scholar
Jaeckel, L. A. (1971), ‘Robust estimates of location: symmetric and asymmetric contamination’, Ann. Math. Stat. 42, 1020–1035.Google Scholar
Kallianpur, G. (1963), ‘Von Mises functions and maximum likelihood estimation’, Sankhya Ser. A 23, 149–158.Google Scholar
Malmquist, S. (1950), ‘On a property of order statistics from a rectangular distribution’, Skand. Akt. 33, 214–222.Google Scholar
Reeds, J. A. (1976), On the definition of Von Mises functions, (Ph.D. Thesis, Dept. of Statistics, Harvard Univ., Cambridge).Google Scholar
Rieder, H. (1978), ‘A robust asymptotic testing model’, Ann. Statist. 6, 1080–1094.Google Scholar
Rieder, H. (1980), ‘Estimators derived from robust tests’, Ann. Statist. 8, 106–115.Google Scholar
Serfling, R. J. (1980), Approximation theorems of mathematical statistics, (Wiley, New York).Google Scholar