{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,16]],"date-time":"2026-09-16T11:02:44Z","timestamp":1789556564287,"version":"build-2803163510"},"reference-count":36,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T00:00:00Z","timestamp":1770681600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004190","name":"NSF","doi-asserted-by":"publisher","award":["DGE-1762114"],"award-info":[{"award-number":["DGE-1762114"]}],"id":[{"id":"10.13039\/501100004190","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004190","name":"NSF","doi-asserted-by":"publisher","award":["DMS-1561814"],"award-info":[{"award-number":["DMS-1561814"]}],"id":[{"id":"10.13039\/501100004190","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000050","name":"NHLBI","doi-asserted-by":"publisher","award":["HL137808"],"award-info":[{"award-number":["HL137808"]}],"id":[{"id":"10.13039\/100000050","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100020940","name":"NIH","doi-asserted-by":"publisher","award":["R01-GM114029"],"award-info":[{"award-number":["R01-GM114029"]}],"id":[{"id":"10.13039\/100020940","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,9,16]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>It is often of interest to make inference on an unknown function that is a local parameter of the data-generating mechanism, such as a density or regression function. Such estimands can typically only be estimated at a slower-than-parametric rate in nonparametric and semiparametric models, and performing calibrated inference can be challenging. In many cases, these estimands can be expressed as the minimizer of a population risk functional. Here, we propose a general framework that leverages such representation and provides a nonparametric extension of the score test for inference on an infinite-dimensional risk minimizer. We demonstrate that our framework is applicable in a wide variety of problems. As both analytic and computational examples, we describe how to use our general approach for inference on a mean regression function under (i) nonparametric and (ii) partially additive models, and evaluate the operating characteristics of the resulting procedures via simulations. Assessment of effect heterogeneity, inference on density functions, and conditional independence testing are discussed as additional examples.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkag043","type":"journal-article","created":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T12:26:02Z","timestamp":1768911962000},"page":"1321-1344","source":"Crossref","is-referenced-by-count":1,"title":["Inference on function-valued parameters using a restricted score test"],"prefix":"10.1093","volume":"88","author":[{"given":"Aaron","family":"Hudson","sequence":"first","affiliation":[{"name":"Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Center , Seattle, WA 98109 ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2106-0953","authenticated-orcid":false,"given":"Marco","family":"Carone","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, University of Washington , Seattle, WA 98105 ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8846-3533","authenticated-orcid":false,"given":"Ali","family":"Shojaie","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, University of Washington , Seattle, WA 98105 ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,2,10]]},"reference":[{"key":"2026091606113291900_qkag043-B1","author":"Barron","year":"1989"},{"key":"2026091606113291900_qkag043-B2","author":"Benkeser","year":"2016"},{"key":"2026091606113291900_qkag043-B3","volume-title":"Efficient and adaptive estimation for semiparametric models","author":"Bickel","year":"1998"},{"issue":"522","key":"2026091606113291900_qkag043-B4","doi-asserted-by":"publisher","first-page":"767","DOI":"10.1080\/01621459.2017.1285776","article-title":"On the effect of bias estimation on coverage accuracy in nonparametric inference","volume":"113","author":"Calonico","year":"2018","journal-title":"Journal of the American Statistical Association"},{"key":"2026091606113291900_qkag043-B5","doi-asserted-by":"publisher","author":"Calonico","year":"2019","DOI":"10.18637\/jss.v091.i08"},{"issue":"1","key":"2026091606113291900_qkag043-B6","doi-asserted-by":"publisher","first-page":"C1","DOI":"10.1111\/ectj.12097","article-title":"Double\/debiased machine learning for treatment and structural parameters","volume":"21","author":"Chernozhukov","year":"2018","journal-title":"The Econometrics Journal"},{"key":"2026091606113291900_qkag043-B7","doi-asserted-by":"publisher","first-page":"1564","DOI":"10.1214\/14-AOS1230","article-title":"Gaussian approximation of suprema of empirical processes","volume":"42","author":"Chernozhukov","year":"2014","journal-title":"Annals of Statistics"},{"issue":"424","key":"2026091606113291900_qkag043-B8","doi-asserted-by":"publisher","first-page":"1287","DOI":"10.1080\/01621459.1993.10476410","article-title":"Confidence bands in nonparametric regression","volume":"88","author":"Eubank","year":"1993","journal-title":"Journal of the American Statistical Association"},{"key":"2026091606113291900_qkag043-B9","first-page":"1","article-title":"A general framework for the analysis of kernel-based tests","volume":"25","author":"Fern\u00e1ndez","year":"2024","journal-title":"Journal of Machine Learning Research"},{"key":"2026091606113291900_qkag043-B10","doi-asserted-by":"publisher","author":"Fu","year":"2017","DOI":"10.18637\/jss.v094.i14"},{"issue":"2","key":"2026091606113291900_qkag043-B11","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1016\/j.jmva.2011.11.004","article-title":"Asymptotic normality of support vector machine variants and other regularized kernel methods","volume":"106","author":"Hable","year":"2012","journal-title":"Journal of Multivariate Analysis"},{"issue":"2","key":"2026091606113291900_qkag043-B12","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1080\/02331889108802305","article-title":"Edgeworth expansions for nonparametric density estimators, with applications","volume":"22","author":"Hall","year":"1991","journal-title":"Statistics"},{"key":"2026091606113291900_qkag043-B13","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1214\/aos\/1176348651","article-title":"Effect of bias estimation on coverage accuracy of bootstrap confidence intervals for a probability density","volume":"20","author":"Hall","year":"1992","journal-title":"Annals of Statistics"},{"issue":"4","key":"2026091606113291900_qkag043-B14","doi-asserted-by":"publisher","first-page":"1892","DOI":"10.1214\/13-AOS1137","article-title":"A simple bootstrap method for constructing nonparametric confidence bands for functions","volume":"41","author":"Hall","year":"2013","journal-title":"Annals of Statistics"},{"issue":"2","key":"2026091606113291900_qkag043-B15","doi-asserted-by":"publisher","first-page":"778","DOI":"10.1214\/aos\/1176348120","article-title":"Bootstrap simultaneous error bars for nonparametric regression","volume":"19","author":"Hardle","year":"1991","journal-title":"Annals of Statistics"},{"key":"2026091606113291900_qkag043-B16","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-84858-7","volume-title":"The elements of statistical learning: Data mining, inference, and prediction","author":"Hastie","year":"2009"},{"issue":"3","key":"2026091606113291900_qkag043-B17","doi-asserted-by":"publisher","first-page":"292","DOI":"10.1080\/00031305.2021.2021984","article-title":"Demystifying statistical learning based on efficient influence functions","volume":"76","author":"Hines","year":"2022","journal-title":"The American Statistician"},{"issue":"1","key":"2026091606113291900_qkag043-B18","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1016\/S0304-4076(02)00157-4","article-title":"Disease cases and their medical costs attributable to smoking: An analysis of the national medical expenditure survey","volume":"112","author":"Johnson","year":"2003","journal-title":"Journal of Econometrics"},{"issue":"532","key":"2026091606113291900_qkag043-B19","doi-asserted-by":"publisher","first-page":"2084","DOI":"10.1080\/01621459.2019.1689984","article-title":"Kernel meets sieve: Post-regularization confidence bands for sparse additive model","volume":"115","author":"Lu","year":"2020","journal-title":"Journal of the American Statistical Association"},{"issue":"4","key":"2026091606113291900_qkag043-B20","doi-asserted-by":"publisher","first-page":"1534","DOI":"10.1214\/24-AOS2403","article-title":"One-step estimation of differentiable Hilbert-valued parameters","volume":"52","author":"Luedtke","year":"2024","journal-title":"Annals of Statistics"},{"issue":"6","key":"2026091606113291900_qkag043-B21","doi-asserted-by":"publisher","first-page":"1937","DOI":"10.1214\/aos\/1034713641","article-title":"Automatic bandwidth choice and confidence intervals in nonparametric regression","volume":"23","author":"Neumann","year":"1995","journal-title":"Annals of Statistics"},{"key":"2026091606113291900_qkag043-B22","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-5769-1","volume-title":"Contributions to a general asymptotic statistical theory","author":"Pfanzagl","year":"1982"},{"key":"2026091606113291900_qkag043-B23","author":"Rao","year":"1948"},{"issue":"4","key":"2026091606113291900_qkag043-B24","doi-asserted-by":"publisher","first-page":"931","DOI":"10.2307\/1912705","article-title":"Root-n-consistent semiparametric regression","volume":"56","author":"Robinson","year":"1988","journal-title":"Econometrica: Journal of the Econometric Society"},{"key":"2026091606113291900_qkag043-B25","author":"Shah","year":"2020"},{"key":"2026091606113291900_qkag043-B26","doi-asserted-by":"publisher","first-page":"1328","DOI":"10.1214\/aos\/1176325631","article-title":"Simultaneous confidence bands for linear regression and smoothing","volume":"22","author":"Sun","year":"1994","journal-title":"Annals of Statistics"},{"issue":"1","key":"2026091606113291900_qkag043-B27","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1093\/jrsssb\/qkae041","article-title":"Debiased inference for a covariate-adjusted regression function","volume":"87","author":"Takatsu","year":"2025","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"key":"2026091606113291900_qkag043-B28","volume-title":"Empirical processes in M-estimation","author":"van de Geer","year":"2000"},{"key":"2026091606113291900_qkag043-B29","author":"van der Laan","year":"2018"},{"issue":"1","key":"2026091606113291900_qkag043-B30","doi-asserted-by":"publisher","first-page":"Article 25","DOI":"10.2202\/1544-6115.1309","article-title":"Super learner","volume":"6","author":"van der Laan","year":"2007","journal-title":"Statistical Applications in Genetics and Molecular Biology"},{"key":"2026091606113291900_qkag043-B31","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-21700-0","volume-title":"Unified methods for censored longitudinal data and causality","author":"van der Laan","year":"2003"},{"key":"2026091606113291900_qkag043-B32","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-9782-1","volume-title":"Targeted learning: Causal inference for observational and experimental data","author":"van der Laan","year":"2011"},{"key":"2026091606113291900_qkag043-B33","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2545-2","volume-title":"Weak convergence and empirical processes","author":"van der Vaart","year":"1996"},{"key":"2026091606113291900_qkag043-B34","volume-title":"Asymptotic statistics","author":"van der Vaart","year":"2000"},{"issue":"3","key":"2026091606113291900_qkag043-B35","doi-asserted-by":"publisher","first-page":"657","DOI":"10.1111\/rssb.12504","article-title":"Assumption-lean inference for generalised linear model parameters","volume":"84","author":"Vansteelandt","year":"2020","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"key":"2026091606113291900_qkag043-B36","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970128","volume-title":"Spline models for observational data","author":"Wahba","year":"1990"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/88\/4\/1321\/66837392\/qkag043.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/88\/4\/1321\/66837392\/qkag043.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,16]],"date-time":"2026-09-16T10:11:39Z","timestamp":1789553499000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/88\/4\/1321\/8470849"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,10]]},"references-count":36,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2026,2,10]]},"published-print":{"date-parts":[[2026,9,16]]}},"URL":"https:\/\/doi.org\/10.1093\/jrsssb\/qkag043","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,9]]},"published":{"date-parts":[[2026,2,10]]}}}