On conditions for asymptotic normality of Fisher's one-step estimators in one-parameter families of distributions

We consider asymptotic behavior of one-step statistical estimators introduced by R. Fisher as approximations for consistent maximum likelihood estimators. Some sufficient conditions are found for these one-step estimators to be asymptotically normal even in the cases when either the maximum likelihood estimators may not exist or exist but be inconsistent. Investigated are connections between the smoothness conditions for the density of the sample distribution and the rate of proximity of the preliminary estimator and the parameter which are needed for fulfillment of the properties under considerations.