Robust estimation in the linear model with asymmetric error distributions

In the linear model Xn - 1 = Cn - p[theta]p - 1 + En - 1, Huber's theory of robust estimation of the regression vector [theta]p - 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 [var epsilon], 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 [theta]p - 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.