Local variable selection of nonlinear nonparametric systems by first order expansion

Local variable selection by first order expansion for nonlinear nonparametric systems is investigated in the paper. By substantially modifying the algorithms developed in our earlier work (Bai et al., 2014), the previous results have been considerably strengthened under much less restrictive conditions. Firstly, the estimates generated by the modified algorithms are shown to have both the set and parameter convergence with probability one, rather than only the set convergence in probability given in our earlier work. Secondly, several technical assumptions, e.g., the lower and upper bounds on the growth of some random sequences, which practically are uncheckable, have been removed. Thirdly, not only the consistency but also the convergence rate of estimates have been established. Besides, a generalization of the proposed algorithms is also introduced.