## The Annals of Statistics

### On the Minimax Property for $R$-Estimators of Location

John R. Collins

#### Abstract

Consider the problem of estimating the unknown location parameter $\theta$ based on a random sample from $F(x - \theta)$, where $F$ is an unknown member of the class of distribution functions $\mathscr{F} = \{F: F$ is symmetric about 0 and $\sup_x |F(x) - \Phi(x)| \leq \varepsilon\}$, where $\Phi$ denotes the standard normal distribution function. Huber (1964) showed that $M$-estimation has a minimax property for this model, whereas Sacks and Ylvisaker (1972) showed that $L$-estimation fails to have the minimax property. It is shown here that $R$-estimation does have the minimax property for this model.

#### Article information

Source
Ann. Statist., Volume 11, Number 4 (1983), 1190-1195.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176346331

Digital Object Identifier
doi:10.1214/aos/1176346331

Mathematical Reviews number (MathSciNet)
MR720263

Zentralblatt MATH identifier
0542.62024

JSTOR
links.jstor.org

Subjects
Primary: 62G35: Robustness
Secondary: 62G05: Estimation

#### Citation

Collins, John R. On the Minimax Property for $R$-Estimators of Location. Ann. Statist. 11 (1983), no. 4, 1190--1195. doi:10.1214/aos/1176346331. https://projecteuclid.org/euclid.aos/1176346331