Second order asymptotic relations of M-estimators and R-estimators in two-sample location model

Let T_M be an M-estimator (maximum likelihood type estimator) and T_R be an R-estimator (Hodges-Lehmann's estimator) of the shift parameter Δ in the two-sample location model. The asymptotic representation of √N(T_M-T_R) up to a term of the order O_p(N^{-$\text{1}\text{4}$}) is derived which is valid if the functions Ψ and ? generating T_M and T_R, respectively, decompose into an absolutely continuous and a step-function components; the order O_p(N^{-$\text{1}\text{4}$}) cannot be improved unless the discontinuous components vanish. As a consequence, the conditions under which √N(T_M-T_R)=O_p(N^{-$\text{1}\text{4}$}) are obtained. The main tool for obtaining the results is the second order asymptotic linearity of the pertaining linear rank statistics which is proved here under the assumption that the score-generating function ? has some jump-discontinuities.