On consistency of redescending M-kernel smoothers

M-estimators and M-kernel estimators with a redescending ψ-function are not in general consistent. This is often handled by means of coupling the estimator to a consistent one. Coupling the estimator to the (inconsistent) starting point improves the jump preserving properties. However, the consistency depends heavily on the shape of the density of the residuals. This paper shows inconsistency under convenient conditions as well as consistency – even at jump points – under somewhat stronger conditions. Research supported by the Friedrich Ebert Foundation and by grant Mu 1031/4-1/2 of the Deutsche Forschungsgemeinschaft