Adaptive estimation of functionals in nonparametric instrumental regression

TL;DR

This paper introduces an adaptive, data-driven estimator for nonparametric instrumental regression that achieves near-minimax optimal convergence rates by combining dimension reduction, thresholding, and model selection techniques.

Contribution

It proposes a novel adaptive plug-in estimator that automatically selects the dimension parameter, improving estimation accuracy in nonparametric instrumental variable models.

Findings

Estimator is consistent and nearly minimax optimal.

Adaptive method effectively chooses the dimension parameter m.

The approach applies to classical smoothness and specific examples like pointwise estimation.

Abstract

We consider the problem of estimating the value l({\phi}) of a linear functional, where the structural function {\phi} models a nonparametric relationship in presence of instrumental variables. We propose a plug-in estimator which is based on a dimension reduction technique and additional thresholding. It is shown that this estimator is consistent and can attain the minimax optimal rate of convergence under additional regularity conditions. This, however, requires an optimal choice of the dimension parameter m depending on certain characteristics of the structural function {\phi} and the joint distribution of the regressor and the instrument, which are unknown in practice. We propose a fully data driven choice of m which combines model selection and Lepski's method. We show that the adaptive estimator attains the optimal rate of convergence up to a logarithmic factor. The theory in this paper is illustrated by considering classical smoothness assumptions and we discuss examples such as pointwise estimation or estimation of averages of the structural function {\phi}.