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These nuisance functions are often weakly identified, meaning the moment restrictions are ill-posed and may admit multiple solutions. This paper proposes a novel condition for the functional to be strongly identified (amenable to n) rate asymptotically normal estimation) even when the nuisance function remains weakly identified. The condition implies the existence of debiasing nuisance functions. We propose penalized minimax estimators for both the primary and debiasing nuisance functions. These estimators accommodate flexible function classes and, crucially, converge to fixed limits determined by the penalization, irrespective of the nuisances' identifiability. We use these penalized estimators to construct a debiased functional estimator and prove its asymptotic normality under general high-level conditions, leading to valid confidence intervals. Our method is illustrated in partially linear proximal causal inference and instrumental variable regression problems.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkaf075","type":"journal-article","created":{"date-parts":[[2025,11,16]],"date-time":"2025-11-16T12:54:52Z","timestamp":1763297692000},"page":"998-1028","source":"Crossref","is-referenced-by-count":0,"title":["Inference on strongly identified functionals of weakly identified functions"],"prefix":"10.1093","volume":"88","author":[{"given":"Andrew","family":"Bennett","sequence":"first","affiliation":[{"name":"Morgan Stanley, New York, NY 10036 ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nathan","family":"Kallus","sequence":"additional","affiliation":[{"name":"Cornell Tech, Cornell University , New York, NY 10044 ,","place":["USA"]},{"name":"Netflix , New York, NY 10003 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