Abstract: | In the linear model Xn × 1 = Cn × pθp × 1 + En × 1, Huber's theory of robust estimation of the regression vector θp × 1 is adapted for two models for the partially specified common distribution F of the i.i.d. components of the error vector En × 1. In the first model considered, the restriction of F to a set −a0, b0] is a standard normal distribution contaminated, with probability , by an unknown distribution symmetric about 0. In the second model, the restriction of F to −a0, b0] is completely specified (and perhaps asymmetrical). In both models, the distribution of F outside the set −a0, b0] is completely unspecified. For both models, consistent and asymptotically normal M-estimators of θp × 1 are constructed, under mild regularity conditions on the sequence of design matrices {Cn × p}. Also, in both models, M-estimators are found which minimize the maximal mean-squared error. The optimal M-estimators have influence curves which vanish off compact sets. |