  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.
