Journal of the Australian Mathematical Society (Series A)

Research Article

Statistical expansions and locally uniform Fréchet differentiability

T. Bednarskia1, B. R. Clarkea2 and W. Kolkiewicza3

a1 Institute of Mathematics Polish Academy of Sciences Warsaw, Poland

a2 Institute of Mathematics Polish Academy of Sciences Warsaw, Poland

a3 Murdoch University Murdoch, WA 6150, Australia

Abstract

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.

(Received June 23 1989)

(Revised November 09 1989)

1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision)

  • primary 62 E 20;
  • secondary 62 G 35

Keywords and phrases

  • infinitesimal neighbourhoods;
  • Fréchet differentiability;
  • strong expansions;
  • M-functionals