Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T20:37:24.223Z Has data issue: false hasContentIssue false
  • English
  • Français

A remark on robustness and weak continuity of M-estimators

Part of: Distribution theory Nonparametric inference

Published online by Cambridge University Press:  09 April 2009

Brenton R. Clarke
Brenton R. Clarke
Affiliation:
Mathematics and Statistics Division of Science and Engineering Murdoch UniversityMurdoch, WA 6150Australia e-mail: clarke@prodigal.murdoch.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Global weak continuity of M-functionals in a neighbourhood of the parametric distribution is established. This has implications for robustness of M-estimators vis a vis definitions put forward by Hampel. For instance the Tukey bisquare location estimator is robust on neighbourhoods of the parametric model, but the median is not.

Keywords

M-functionalsProhorov metric distanceweak continuity

MSC classification

Secondary: 62G35: Robustness 62E20: Asymptotic distribution theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 3 , June 2000 , pp. 411 - 418
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Beaton, A. E. and Tukey, J. W., ‘The fitting of power series meaning polynomials, illustrated on band-spectroscopic data’, Technometrics 16 (1974), 147–185.CrossRefGoogle Scholar
[2]Bednarski, T. and Zontek, S., ‘Robust estimation of parameters in a mixed unbalanced model’, Ann. Statist. 24 (1996), 1493–1510.CrossRefGoogle Scholar
[3]Clarke, B. R., ‘Uniqueness and Fréchet differentiability of functional solutions to maximum likelihood type equations’, Ann. Statist. 11 (1983), 1196–1205.CrossRefGoogle Scholar
[4]Clarke, B. R., ‘The selection functional’, Probab. Math. Statist. 11 (1991), 149–156.Google Scholar
[5]Clarke, B. R. and Heathcote, C. R., ‘Robust estimation of k-component univariate normal mixtures’, Ann. Inst. Statist. Math. 46 (1994), 83–93.CrossRefGoogle Scholar
[6]Hampel, F. R., ‘A general qualitative definition of robustness’, Ann. Math. Statist. 42 (1971), 1887–1896.CrossRefGoogle Scholar
[7]Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A., Robust statistics: the approach based on influence functions (Wiley, New York, 1986).Google Scholar
[8]Heathcote, C. R. and Silvapulle, M. J., ‘Minimum mean squared estimation of location and scale parameters under misspecification of the model’, Biometrika 68 (1981), 501–514.CrossRefGoogle Scholar
[9]Huggins, R. M., ‘On the robust analysis of variance components models for pedigree data’, Austral. J. Statist. 35 (1993), 43–57.CrossRefGoogle Scholar
[10]Prohorov, Y. V., ‘Convergence of random processes and limit theorems in probability’, Theory Probab. Appl. 1 (1956), 157–214.CrossRefGoogle Scholar
[11]Varadarajan, V. S., ‘On the convergence of probability distributions’, Sankhyā 19 (1958), 23–26.Google Scholar
[12]von Mises, R., ‘On the asymptotic distribution of differentiable statistical functions’, Ann. Math. Statist. 18 (1947), 309–348.CrossRefGoogle Scholar