Robust M ‐estimators of scale: Minimax bias versus maximal variance

In the location‐scale estimation problem, we study robustness properties of M ‐estimators of the scale parameter under unknown ϵ‐contamination of a fixed symmetric unimodal error distribution F 0 . Within a general class of M ‐estimators, the estimator with minimax asymptotic bias is shown to lie within the subclass of α‐interquantile ranges of the empirical distribution symmetrized about the sample median. Our main result is that as ϵ → 0, the limiting minimax asymptotic bias estimator is sometimes (e.g., when F o is Cauchy), but not always, the median absolute deviation about the median. It is also shown that contamination in the neighbourhood of a discontinuity of the influence function of a minimax bias estimator can sometimes inflate the asymptotic variance beyond that achieved by placing all the ϵ‐contamination at infinity. This effect is quantified by a new notion of asymptotic efficiency that takes into account the effect of infinitesimal contamination of the parametric model for the error distribution.