Efficiency and Robustness of a Resampling M-Estimator in the Linear Model

In the literature, there are basically two kinds of resampling methods for least squares estimation in linear models; the E-type (the efficient ones like the classical bootstrap), which is more efficient when error variables are homogeneous, and the R-type (the robust ones like the jackknife), which is more robust for heterogeneous errors. However, for M-estimation of a linear model, we find a counterexample showing that a usually E-type method is less efficient than an R-type method when error variables are homogeneous. In this paper, we give sufficient conditions under which the classification of the two types of the resampling methods is still true.