## The Annals of Statistics

### $L$- and $R$-Estimation and the Minimax Property

#### Abstract

Let $\{X_i\}$ be a sample from $F(x - \theta)$ where $F$ is in a class $\mathscr{F}$ of symmetric distributions on the line and $\theta$ is the location parameter to be estimated. Huber has shown that maximum likelihood estimation has a minimax property over a convex $\mathscr{F}$. Here a simple convex $\mathscr{F}$ is given for which neither $L$- nor $R$-estimation has the minimax property. In particular, this example shows that a recent assertion concerning $L$-estimation is not true.

#### Article information

Source
Ann. Statist., Volume 10, Number 2 (1982), 643-645.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176345808

Digital Object Identifier
doi:10.1214/aos/1176345808

Mathematical Reviews number (MathSciNet)
MR653542

Zentralblatt MATH identifier
0488.62022

JSTOR
Sacks, Jerome; Ylvisaker, Donald. $L$- and $R$-Estimation and the Minimax Property. Ann. Statist. 10 (1982), no. 2, 643--645. doi:10.1214/aos/1176345808. https://projecteuclid.org/euclid.aos/1176345808