Refinement of Fisher's One-Step Estimators in the Case of Slowly Converging Initial Estimators

We discuss a construction of asymptotically efficient one-step estimators in the case when there exist initial $n^{\beta}$-consistent estimators. The well-known one-step estimators proposed by R. Fisher to approximate consistent maximum likelihood estimators allow us to construct one-step asymptotically efficient estimators only in the case when $\beta\ge 1/4$. The estimators proposed in the present paper transform $n^{\beta}$-consistent estimators for $\beta< 1/4$ to asymptotically optimal ones using only one iteration of a Newton-type approximation. In the present paper, sufficient conditions are found for the one-step estimators to be asymptotically efficient even in the case when the corresponding maximum likelihood estimators either do not exist or are not consistent.