A multidimensional Newton-Raphson method and its applications to the existence of asymptotic Fn-estimators and their stochastic expansions

Given a suitable function F_n we define a class of estimators called asymptotic F_n-estimators (i.e., estimators which approximate the solution of F_n(θ) = 0). It is proved that this class is nonvoid if appropriate regularity conditions are fulfilled and if one has at hand a suitable initial estimator. Furthermore, it is shown that F_n-estimators admit a stochastic expansion (which enables to give results on asymptotic expansions for the distribution of these estimators).