Efficiency of Z-estimators indexed by the objective functions

We study the convergence of $Z$-estimators $\widehat \theta(\eta)\in \mathbb R^p$forwhichtheobjectivefunctiondependsonaparameter$\eta$ that belongs to a Banach space $\mathcal{H}$. Our results include the uniform consistency over $\mathcal{H}$ and the weak convergence in the space of bounded ${\mathbb{R}}^p$-valued functions defined on $\mathcal{H}$. Furthermore when $\eta$ is a tuning parameter optimally selected at ${\eta }_0$, we provide conditions under which an estimated $\widehat{\eta }$ can be replaced by ${\eta }_0$ without affecting the asymptotic variance. Interestingly, these conditions are free from any rate of convergence of $\widehat{\eta }$ to ${\eta }_0$ but they require the space described by $\widehat{\eta }$ to be not too large. We highlight several applications of our results and we study in detail the case where $\eta$ is the weight function in weighted regression.