{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19350",
  "author_total": 2,
  "authors": [
    {
      "name": "Ellen Graham"
    },
    {
      "name": "Andrea Rotnitzky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cb29fv1zq5s11.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19350",
  "fulltext_url": null,
  "id": "cb29fv1zq5s11",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19350",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Advanced Causal Inference Techniques"
    ]
  ],
  "openalex": "W7222284511",
  "pdf_url": "https://arxiv.org/pdf/2610.19350",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cb29fv1zq5s11.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19350",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19350"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "On Local Inference for Functionals of Ill-Posed Inverse Problems",
  "title_text": "On Local Inference for Functionals of Ill-Posed Inverse Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:42Z",
  "url": "https://research.draconitic.com/paper/cb29fv1zq5s11",
  "venue": "arXiv",
  "versions": []
}