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{1176345808,
author = {Sacks, Jerome and Ylvisaker, Donald},
title = {$L$- and $R$-Estimation and the Minimax Property},
journal = {Ann. Statist.},
volume = {10},
number = {1},
year = {1982},
pages = { 643-645},
language = {en},
url = {http://dml.mathdoc.fr/item/1176345808}
}
Sacks, Jerome; Ylvisaker, Donald. $L$- and $R$-Estimation and the Minimax Property. Ann. Statist., Tome 10 (1982) no. 1, pp. 643-645. http://gdmltest.u-ga.fr/item/1176345808/