Robust estimation in the linear model with asymmetric error distributions

Abstract

In the linear model X n × 1 = C n × p θ p × 1 + E n × 1 , Huber's theory of robust estimation of the regression vector θ p × 1 is adapted for two models for the partially specified common distribution F of the i.i.d. components of the error vector E n × 1 . In the first model considered, the restriction of F to a set [− a 0, b 0] is a standard normal distribution contaminated, with probability ε, by an unknown distribution symmetric about 0. In the second model, the restriction of F to [− a 0, b 0] is completely specified (and perhaps asymmetrical). In both models, the distribution of F outside the set [− a 0, b 0] is completely unspecified. For both models, consistent and asymptotically normal M-estimators of θ p × 1 are constructed, under mild regularity conditions on the sequence of design matrices { C n × 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.