On Local Inference for Functionals of Ill-Posed Inverse Problems

Abstract

We study inference for continuous linear functionals of solutions to ill-posed inverse problems, including Euclidean parameters arising in nonparametric instrumental variables, proximal causal inference, and shadow-variable models. Although pointwise root-$n$ inference is possible under standard conditions, we show that this need not translate into locally reliable inference. Under a model imposing only minimal regularity conditions, the target is Hellinger discontinuous when both inverse problems are ill-posed: arbitrarily small changes in the data-generating law can produce arbitrarily large changes in the target, ruling out informative locally uniform inference. Imposing a uniform bound on a minimum-norm solution restores Hellinger continuity under regularity conditions, but does not by itself ensure stable root-n inference. If, in addition, both inverse problems are ill-posed and the solution whose norm is uniformly bounded is nonunique at a law $P_0$, then under regularity conditions the target fails to be pathwise differentiable; consequently, asymptotically linear estimators exhibit nonvanishing root-$n$ bias under some local alternatives, causing Wald confidence intervals to undercover asymptotically. Under regularity conditions, pathwise differentiability is restored if the uniformly bounded solution is unique at $P_0$. It is also restored, regardless of uniqueness, under the stronger requirement that the corresponding solution satisfy a uniformly bounded $β$-source condition for some $β$>0. We relate these regimes to very weak, weak, and strong identification in instrumental variables models.