Asymptotic relations between L- and M-estimators in the linear model

We obtain Bahadur-type representations for one-step L-estimators, M- and one-step M-estimators in the linear model. The order of the remainder terms in these representations depends on the smooth-ness of the weight function for L-estimators and on the smoothness of the ψ-function for M- and one-step M-estimators. We use the representations to investigate the asymptotic relations between these estimators. In particular, we show that asymptotically equivalent L- and M-estimators of the slope parameter exist even when the underlying distribution is asymmetric. It is important to consider the asymmetric case for both practical and robustness reasons: first, there is no compelling argument which precludes asymmetric distributions from arising in practice, and, secondly, even if a symmetric model can be posited, it is important to allow for the possibility of mild (and therefore difficult to detect) departures from the symmetric model.