On the convergence of Newton's method when estimating higher dimensional parameters

In this paper, we consider the estimation of a parameter of interest where the estimator is one of the possibly several solutions of a set of nonlinear empirical equations. Since Newton's method is often used in such a setting to obtain a solution, it is important to know whether the so obtained iteration converges to the locally unique consistent root to the aforementioned parameter of interest. Under some conditions, we show that this is eventually the case when starting the iteration from within a ball about the true parameter whose size does not depend on n. Any preliminary almost surely consistent estimate will eventually lie in such a ball and therefore provides a suitable starting point for large enough n. As examples, we will apply our results in the context of M-estimates, kernel density estimates, as well as minimum distance estimates.