Simple Adaptive Estimation of Quadratic Functionals in Nonparametric IV Models

TL;DR

This paper develops an adaptive estimator for quadratic functionals in nonparametric IV models, achieving minimax optimal rates across different ill-posedness scenarios without prior knowledge of smoothness or ill-posedness.

Contribution

It introduces a data-driven, Lepski-type method for selecting the sieve dimension, enabling adaptive minimax estimation in complex NPIV models.

Findings

Achieves the minimax convergence rate matching the lower bound.

Adaptive estimator performs well in severely and mildly ill-posed cases.

Attains near-optimal rates up to a logarithmic factor in irregular cases.

Abstract

This paper considers adaptive, minimax estimation of a quadratic functional in a nonparametric instrumental variables (NPIV) model, which is an important problem in optimal estimation of a nonlinear functional of an ill-posed inverse regression with an unknown operator. We first show that a leave-one-out, sieve NPIV estimator of the quadratic functional can attain a convergence rate that coincides with the lower bound previously derived in Chen and Christensen [2018]. The minimax rate is achieved by the optimal choice of the sieve dimension (a key tuning parameter) that depends on the smoothness of the NPIV function and the degree of ill-posedness, both are unknown in practice. We next propose a Lepski-type data-driven choice of the key sieve dimension adaptive to the unknown NPIV model features. The adaptive estimator of the quadratic functional is shown to attain the minimax optimal rate in the severely ill-posed case and in the regular mildly ill-posed case, but up to a multiplicative $\sqrt{\log n}$ factor in the irregular mildly ill-posed case.