Unimodality and the asymptotics of M-estimators

Unimodality, in its weaker and stronger forms, enters the robustness investigations somehow less often than symmetry. We point out how unimodality affects the asymptotics of M-estimators under heterogeneous ("non-i.i.d.") errors. Sufficient conditions are given for consistency, with rates, of NI-estimators in unimodal heterogeneous location models. For heteroscedastic models, a particular case of heterogeneous ones, a necessary and sufficient consistency condition, with rates, is provided for the L-1 estimator - the sample median.