Bias-robust L-estimators of a scale parameter

We derive optimal bias-robust L-estimators of a scale parameter a based on random samples from F(((.)-theta)/sigma), where theta and sigma are unknown and F is an unknown member of a epsilon-contaminated neighborhood of a fixed symmetric error distribution F-0. Within a very general class S of L-estimators which are Fisher-consistent at F0, we solve for: (i) the estimator with minimax asymptotic bias over the epsilon-contamination neighborhood; and (ii) the estimator with minimum gross error sensitivity at F-0 [the limiting case of (i) as epsilon --> 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.