On the asymptotics of Z-estimators indexed by the objective functions

We study the convergence of Z-estimators theta(eta) is an element of R-p for which the objective function depends on a parameter. that belongs to a Banach space H. Our results include the uniform consistency over H and the weak convergence in the space of bounded R-p-valued functions defined on H. When. is a tuning parameter optimally selected at eta(0), we provide conditions under which eta(0) can be replaced by an estimated eta without affecting the asymptotic variance. Interestingly, these conditions are free from any rate of convergence of eta to eta(0) but require the space described by eta to be not too large in terms of bracketing metric entropy. In particular, we show that Nadaraya-Watson estimators satisfy this entropy condition. We highlight several applications of our results and we study the case where. is the weight function in weighted regression.