On rate optimal local estimation in nonparametric instrumental regression

TL;DR

This paper develops a minimax optimal estimator for linear functionals in nonparametric instrumental regression, utilizing dimension reduction and thresholding, with theoretical convergence rates derived under smoothness assumptions.

Contribution

It introduces a new estimator that achieves optimal convergence rates for linear functional estimation in nonparametric instrumental regression, linking structural function smoothness to the conditional expectation operator.

Findings

Estimator achieves minimax optimal convergence rates.

Results apply under classical smoothness assumptions.

Theoretical analysis links function smoothness to the conditional expectation operator.

Abstract

We consider the problem of estimating the value of a linear functional in nonparametric instrumental regression, where in the presence of an instrument W a response Y is modeled in dependence of an endogenous explanatory variable Z. The proposed estimator is based on dimension reduction and additional thresholding. The minimax optimal rate of convergence of the estimator is derived assuming that the structural function and the representer of the linear functional belong to some ellipsoids which are in a certain sense linked to the conditional expectation operator of Z given W. We illustrate these results by considering classical smoothness assumptions.