## The Annals of Statistics

### Robust Estimation in Models for Independent Non-Identically Distributed Data

Rudolf Beran

#### Abstract

This paper concerns robust estimation of the parameter $\theta$ which indexes a parametric model for independent non-identically distributed data. For reasonable choices of contamination neighborhood and of what is to be estimated when the parametric model does not hold, we characterize asymptotically minimax robust estimates of $\theta$. When applied to the normal regression model, the theory yields recipes for the influence curves of optimal robust regression and scale estimates. The contamination neighborhood does not assume regression plus error structure, the regression and scale parameters are estimated simultaneously, and the theory establishes roles for estimates with redescending influence curves as well as for those with monotone influence curves. When applied to the logit and probit models, the theory recommends influence curves which differ markedly from those of the maximum likelihood estimates except in the i.i.d. case.

#### Article information

Source
Ann. Statist., Volume 10, Number 2 (1982), 415-428.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176345783

Digital Object Identifier
doi:10.1214/aos/1176345783

Mathematical Reviews number (MathSciNet)
MR653517

Zentralblatt MATH identifier
0513.62041

JSTOR

Subjects
Primary: 62F35: Robustness and adaptive procedures
Secondary: 62C20: Minimax procedures

#### Citation

Beran, Rudolf. Robust Estimation in Models for Independent Non-Identically Distributed Data. Ann. Statist. 10 (1982), no. 2, 415--428. doi:10.1214/aos/1176345783. https://projecteuclid.org/euclid.aos/1176345783