Maximal asymptotic biases of M-estimators of location with preliminary scale estimates

We consider the problem of finding the maximal asymptotic bias of an M-estimator of a location parameter, using a preliminary estimate of the unknown scale parameter, when the error distribution is assumed to lie in an epsilon-contamination neighborhood of a fixed symmetric unimodal distribution. The least favorable contaminating distribution is shown to put all its mass at infinity under some fairly general conditions. A particular case considered is that of the auxiliary scale estimator being a location-invariant and scale-equivariant version of a trimmed variance.