Weaken the n^{-1/4} rate requirement for nonlinear functionals

Determine whether the standard n^{-1/4} mean-square convergence rate requirement on the machine-learning instrumental variables estimator of the structural function γ for establishing root-n asymptotic normality of debiased estimators of nonlinear functionals m(W, γ) in the nonparametric instrumental variables framework can be weakened, and, if so, characterize alternative conditions under which asymptotic normality holds.

Background

For nonlinear functionals m(W, γ), the paper requires the estimator of γ to achieve a mean-square convergence rate faster than n{-1/4} to control the linearization remainder and secure valid inference. This condition stems from classical semiparametric theory (e.g., Newey, 1994) and is standard in the literature.

However, in ill-posed NPIV problems, attaining such a rate can be stringent, particularly under severe ill-posedness. The authors note that it remains unknown whether the n{-1/4} requirement can be relaxed for nonlinear functionals, raising a fundamental open question about the minimal rate or alternative conditions needed for valid asymptotic normality in this setting.

References

However, as discussed in CNS, it is not known whether it is possible to weaken the $n{-1/4}$ condition for nonlinear functionals, which goes back to .

— Penalized GMM Framework for Inference on Functionals of Nonparametric Instrumental Variable Estimators  (2603.29889 - Bakhitov, 31 Mar 2026) in Section 6, Nonlinear functionals (discussion following Assumption 6)