Conditions of Asymptotic Normality of One-Step M -Estimators

In the case of independent identically distributed observations, we study the asymptotic properties of one-step M -estimators served as explicit approximations of consistent M -estimators. We find rather general conditions for the asymptotic normality of one-step M -estimators. We consider Fisher’s approximations of consistent maximum likelihood estimators and find general conditions guaranteeing the asymptotic normality of the Fisher estimators even in the case where maximum likelihood estimators do not necessarily exist or are not necessarily consistent.