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 P0, 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 P0. 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.
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BibTeX
@misc{graham2026local,
title = {{On Local Inference for Functionals of Ill-Posed Inverse Problems}},
author = {Ellen Graham and Andrea Rotnitzky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19350},
eprint = {2610.19350},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19350},
}
RIS
TY - UNPB
TI - On Local Inference for Functionals of Ill-Posed Inverse Problems
AU - Ellen Graham
AU - Andrea Rotnitzky
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19350
UR - https://arxiv.org/abs/2610.19350
AB - 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.
ER -