Journal of the Australian Mathematical Society (Series A)

Research Article

A remark on robustness and weak continuity of M-estimators

Brenton R. Clarkea1

a1 Mathematics and Statistics Division of Science and Engineering Murdoch University Murdoch, WA 6150 Australia e-mail: clarke@prodigal.murdoch.edu.au

Abstract

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.

(Received January 11 1999)

(Revised November 19 1999)

2000 Mathematics subject classification

  • primary 62G35;
  • secondary 62E20

Keywords and phrases

  • M-functionals;
  • Prohorov metric distance;
  • weak continuity