On efficiency of a class of non-parametric estimates

For a general class of statistics $T_n$ defined by a relation of the form
$$ \sum_{i=1}^n\psi(X_i,T_n)=0, $$
where $X_i$ are observations, a number of results is proved which show that $T_n$ (or, in some cases, their appropriate modifications $T_n^*$) are locally asymptotically minimax estimates of the corresponding functional $\Phi(F)$ of the unknown distribution $F$ provided the family of all admissible distributions $F$ is sufficiently large.