We study asymptotic behavior of one-step $M$-estimators based on samples from arrays of not necessarily identically distributed random variables and representing explicit approximations to the corresponding consistent $M$-estimators. These estimators generalize Fisher's one-step approximations to consistent maximum likelihood estimators. As a consequence, we consider some nonlinear regression problems where the procedure mentioned allow us to construct explicit asymptotically optimal estimators. We also consider the problem of constructing initial estimators which are needed for one-step estimation procedures.