The properties of robust M estimators with randomly right-censored response variables in linear regression models are considered. The most robust and the optimal robust M estimators of the regression parameters are derived within a class of η functions considered in James as well as for a class of η functions corresponding to the general unrestricted class. The usefulness of the estimators corresponding to these two classes are examined. From the computational point of view the James-type η functions are readily obtainable from the η functions in the uncensored case. However, it is found that the breakdown point of the optimal James-type estimators can be lower than the breakdown point of the corresponding optimal robust estimators for nonsymmetric parent distribution functions such as the extreme value distribution. In addition, the efficiency of the optimal James-type estimators is somewhat lower than the efficiency of the optimal robust estimators.
