k-step versions Mn(k)of studentized M-estimator Mnof location generated by various types of score functions are studied. The rate of approximation of Mnby Mn(k)(n → ∞, k fixed) is derived with the aid of the uniform asymptotic linearity results of three-parameters empirical M-processes. While the rate is considerably improved with increasing k for absolutely continuous score functions, it is improved very slowly for functions with jump-discontinuities where it can never be better than Op(n-1).