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

The Annals of Statistics · 1982 · Volume 10 · Issue 2

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.

Keywords

62G20, 62G35, Location parameter estimation, minimax property, robust estimation

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