Efficiency of Z-estimators indexed by the objective functions

TL;DR

This paper investigates the convergence properties of Z-estimators indexed by a parameter in a Banach space, providing uniform consistency, weak convergence results, and conditions for replacing estimated parameters without affecting asymptotic variance, with applications including weighted regression.

Contribution

It offers new theoretical results on the convergence and asymptotic behavior of Z-estimators indexed by parameters in Banach spaces, including conditions for parameter replacement.

Findings

Uniform consistency over the parameter space

Weak convergence in the space of bounded functions

Conditions for replacing estimated parameters without affecting asymptotic variance

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.