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Efficiency of Z-estimators indexed by the objective functions

We study the convergence of $Z$-estimators $\widehat θ(η)\in \mathbb R^p$ for which the objective function depends on a parameter $η$ 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 $η$ is a tuning parameter optimally selected at $η_0$, we provide conditions under which an estimated $\widehat η$ can be replaced by $η_0$ without affecting the asymptotic variance. Interestingly, these conditions are free from any rate of convergence of $\widehat η$ to $η_0$ but they require the space described by $\widehat η$ to be not too large. We highlight several applications of our results and we study in detail the case where $η$ is the weight function in weighted regression.