Simple Adaptive Estimation of Quadratic Functionals in Nonparametric IV Models

Understand

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.