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

摘要

We study the convergence of $Z$-estimators $widehat{heta}(eta)in mathbb{R}^{p}$ 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 $mathcal{H}$. When $eta$ is a tuning parameter optimally selected at $eta_{0}$, we provide conditions under which $eta_{0}$ can be replaced by an estimated $widehat{eta}$ without affecting the asymptotic variance. Interestingly, these conditions are free from any rate of convergence of $widehat{eta}$ to $eta_{0}$ but require the space described by $widehat{eta}$ 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 $eta$ is the weight function in weighted regression.