On the Minimax Property for $R$-Estimators of Location
The Annals of Statistics · 1983 · Volume 11 · Issue 4
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.
Keywords
$R$-estimators, 62G05, 62G35, minimax property, Robust estimation of location
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