Abstract
We obtain Bahadur-type representations for one-stepL-estimators,M- and one-stepM-estimators in the linear model. The order of the remainder terms in these representations depends on the smooth-ness of the weight function forL-estimators and on the smoothness of the ψ-function forM- and one-stepM-estimators. We use the representations to investigate the asymptotic relations between these estimators. In particular, we show that asymptotically equivalentL- andM-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.
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Jurečková, J., Welsh, A.H. Asymptotic relations betweenL- andM-estimators in the linear model. Ann Inst Stat Math 42, 671–698 (1990). https://doi.org/10.1007/BF02481144
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DOI: https://doi.org/10.1007/BF02481144