Asymptotic Relative Efficiencies (ARE) of the robust coordinatewise M-estimators with respect to the robust Maronna-type M-estimators proposed by Singer and Sen (1985) are computed under different elliptically symmetric error distributions. ARE of robust coordinatewise M-estimators with respect to Normal Maximum Likelihood (NML) estimators are also computed under a more general bivariate underlying distribution. The results indicate that the Maronna-type M-estimator is only slightly more efficient than the coordinatewise one, except for extreme cases like the Cauchy or for large dimensionalities where the gain in efficiency may be considerable. Both types of M-estimators perform better than the NML estimators for the proposed departures from normality. For distributions not of the elliptically symmetric type, Maronna-type estimators are not suitable, and hence, the comparison is not of real interest.