The Annals of Statistics

Uniqueness and Frechet Differentiability of Functional Solutions to Maximum Likelihood Type Equations

Brenton R. Clarke

Full-text: Open access

Abstract

Solutions of simultaneous equations of the maximum likelihood type or $M$-estimators can be represented as functionals. Existence and uniqueness of a root in a local region of the parameter space are proved under conditions that are easy to check. If only one root of the equation exists, the resulting statistical functional is Frechet differentiable and robust. When several solutions exists, conditions on the loss criterion used to select the root for the statistic ensure Frechet differentiability. An interesting example of a Frechet differentiable functional is the solution of the maximum likelihood equations for location and scale parameters in a Cauchy distribution. The estimator is robust and asymptotically efficient.

Article information

Source
Ann. Statist., Volume 11, Number 4 (1983), 1196-1205.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176346332

Digital Object Identifier
doi:10.1214/aos/1176346332

Mathematical Reviews number (MathSciNet)
MR720264

Zentralblatt MATH identifier
0541.62023

JSTOR
links.jstor.org

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 62G35: Robustness

Keywords
Statistical functionals weak continuity Frechet differentiability $M$-estimators Cauchy distribution

Citation

Clarke, Brenton R. Uniqueness and Frechet Differentiability of Functional Solutions to Maximum Likelihood Type Equations. Ann. Statist. 11 (1983), no. 4, 1196--1205. doi:10.1214/aos/1176346332. https://projecteuclid.org/euclid.aos/1176346332


Export citation