What is it about?

This paper supplies conditions A which are used to show weak continuity and Frechet differentiability of functional solutions to maximum likelihood type equations, also known as M-estimators, when the defining psi function is continuously differentiable and bounded. It was thought by Huber (1981) "Robust Statistics" Wiley publishers, that..."Unfortunately the concept of Frechet differentiability appears to be too strong in too many cases, the Frechet derivative does not exist, and even if it does, the fact is difficult to establish". The current paper gives conditions which are easily satisfied for M-estimators with bounded and smooth psi - functions. Indeed M-estimators satisfying these conditions are robust.

Featured Image

Why is it important?

The paper makes a break through in understanding of what defines a robust M-estimator. Conditions A have since been used to provide robustness in testing and in general inference. This study supports the intuitive concepts of robustness that were alluded to but not specifically defined in the PhD thesis of Hampel (1968), Berkeley Statistics Department.

Perspectives

This study was the result of my PhD studies at the Australian National University 1977-1980 and it was not until 1986 that I was able to weaken the conditions to include M-estimators with continuous psi - functions but where partial derivatives were not continuous...........I thought of the proof using Conditions A' when I was on a semester Visiting Professor position at the University of North Carolina at Chapel Hill and subsequently wrote the paper at Easter time in 1984 while I was in my first semester of teaching at Murdoch University. All this is subsumed in my 2018 monograph Robustness Theory and Application which is a Wiley Publication

Dr Brenton R. Clarke
Murdoch University

Read the Original

This page is a summary of: Uniqueness and Frechet Differentiability of Functional Solutions to Maximum Likelihood Type Equations, The Annals of Statistics, December 1983, Institute of Mathematical Statistics,
DOI: 10.1214/aos/1176346332.
You can read the full text:

Read

Contributors

The following have contributed to this page