BIAS-ROBUST L -ESTIMATORS OF A SCALE PARAMETER.

We derive optimal bias-robust L -estimators of a scale parameter $\sigma$ based on random samples from $F\lpar \lpar \cdot \!- \theta\rpar /\sigma\rpar$ , where $\theta$ and $\sigma$ are unknown and F is an unknown member of a $\varepsilon$ -contaminated neighborhood of a fixed symmetric error distribution $F_0$ . Within a very general class ${\cal S}$ of L -estimators which are Fisher-consistent at $F_0$ , we solve for: (i) the estimator with minimax asymptotic bias over the $\varepsilon$ -contamination neighborhood; and (ii) the estimator with minimum gross error sensitivity at $F_0$ [the limiting case of (i) as $\varepsilon \rightarrow 0$ ]. The solutions to problems (i) and (ii) are shown, using a generalized method of moment spaces, to be mixtures of at most two interquantile ranges. A graphical method is presented for finding the optimal bias-robust solutions, and examples are given. [ABSTRACT FROM AUTHOR]