Efficiency of Z-estimators indexed by the objective functions

Abstract

We study the convergence of $Z$-estimators $\widehat \theta(\eta)\in \mathbb R^p$ for which the objective function depends on a parameter $\eta$ 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.