The maximum asymptotic bias of S-estimates for regression over the neighborhoods defined by certain special capacities

The maximum asymptotic bias of an S-estimate for regression in the linear model is evaluated over the neighborhoods (called (c,γ)-neighborhoods) defined by certain special capacities, and its lower and upper bounds are derived. As special cases, the (c,γ)-neighborhoods include those in terms of ε-contamination, total variation distance and Riederʹs (ε,δ)-contamination. It is shown that when the model distribution is normal and the (ε,δ)-contamination neighborhood is adopted, the lower and upper bounds of an S-estimate (including the LMS-estimate) based on a jump function coincide with the maximum asymptotic bias. The tables of the maximum asymptotic bias of the LMS-estimate are given. These results are an extension of the corresponding ones due to Martin et al. (Ann. Statist. 17 (1989) 1608), who used ε-contamination neighborhoods.