{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T03:07:07Z","timestamp":1790564827508,"version":"4.1.0"},"reference-count":24,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["UM1 AI068635"],"award-info":[{"award-number":["UM1 AI068635"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["R01HL137808"],"award-info":[{"award-number":["R01HL137808"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["R37 Al029168"],"award-info":[{"award-number":["R37 Al029168"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,10,21]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>This paper examines the construction of confidence sets for parameters defined as linear functionals of a function of $ W $ and $ X $ whose conditional mean given $ Z $ and $ X $ equals the conditional mean of another variable $ Y $ given $ Z $ and $ X $. Many estimands of interest in causal inference can be expressed in this form, including the average treatment effect in proximal causal inference and treatment effect contrasts in instrumental variable models. We derive a necessary condition for a confidence set to be uniformly valid over a model that allows for the dependence between $ W $ and $ Z $ given $ X $ to be arbitrarily weak. We show that, for any such confidence set, there must exist some laws in the model under which, with high probability, the confidence set has a diameter greater than or equal to the diameter of the parameter\u2019s range. In particular, consistent with the weak instrument literature, Wald confidence intervals are not uniformly valid over the aforementioned model when the parameter\u2019s range is infinite. Furthermore, we argue that inverting the score test, a successful approach in that literature, generally fails for the broader class of parameters considered here. We present a method for constructing uniformly valid confidence sets when all variables, but possibly $ Y $, are binary, discuss its limitations and emphasize that developing valid confidence sets for the class of parameters considered here remains an open problem.<\/jats:p>","DOI":"10.1093\/biomet\/asaf067","type":"journal-article","created":{"date-parts":[[2025,9,27]],"date-time":"2025-09-27T11:58:09Z","timestamp":1758974289000},"source":"Crossref","is-referenced-by-count":2,"title":["On the asymptotic validity of confidence sets for linear functionals of solutions to integral equations"],"prefix":"10.1093","volume":"112","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7600-4222","authenticated-orcid":false,"given":"E","family":"Smucler","sequence":"first","affiliation":[{"name":"Glovo , Carrer de Ramon Llull 110 , Barcelona 08005,","place":["Spain"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J M","family":"Robins","sequence":"additional","affiliation":[{"name":"Department of Epidemiology and Biostatistics, Harvard T. H. Chan School of Public Health , 677 Huntington Avenuue , Boston, Massachusetts 02115,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7917-0405","authenticated-orcid":false,"given":"A","family":"Rotnitzky","sequence":"additional","affiliation":[{"name":"Department of Biostatistcs, University of Washington , 3980 15th Avenue NE , Seattle, Washington 98195-1617,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2025,10,6]]},"reference":[{"key":"2026032607095116400_asaf067-B1","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1214\/aoms\/1177730090","article-title":"Estimation of the parameters of a single equation in a complete system of stochastic equations","volume":"20","author":"Anderson","year":"1949","journal-title":"Ann. Math. 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