{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T04:23:46Z","timestamp":1791433426223,"version":"4.3.3"},"reference-count":56,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2025,8,25]],"date-time":"2025-08-25T00:00:00Z","timestamp":1756080000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,2,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>In this paper, we propose a new and unified approach for nonparametric regression and conditional distribution learning. Our approach simultaneously estimates a regression function and a conditional generator using a generative learning framework, where a conditional generator is a function that can generate samples from a conditional distribution. The main idea is to estimate a conditional generator satisfying the constraint that it produces a good regression function estimator. We use deep neural networks to model the conditional generator. Our approach can handle problems with multivariate outcomes and covariates, and can be used to construct prediction intervals. We provide theoretical guarantees by deriving nonasymptotic error bounds and the distributional consistency of our approach under suitable assumptions. We perform numerical experiments to demonstrate the effectiveness and superiority of our approach over some existing approaches in various scenarios.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkaf053","type":"journal-article","created":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T11:43:08Z","timestamp":1754480588000},"page":"330-351","source":"Crossref","is-referenced-by-count":2,"title":["Wasserstein generative regression"],"prefix":"10.1093","volume":"88","author":[{"given":"Shanshan","family":"Song","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, School of Economics and Management, and Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University , Shanghai 200092 ,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tong","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Statistics, The Chinese University of Hong Kong , Hong Kong SAR 999077, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guohao","family":"Shen","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, The Hong Kong Polytechnic University , Hong Kong SAR 999077, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1293-1040","authenticated-orcid":false,"given":"Yuanyuan","family":"Lin","sequence":"additional","affiliation":[{"name":"Department of Statistics, The Chinese University of Hong Kong , Hong Kong SAR 999077, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5218-9269","authenticated-orcid":false,"given":"Jian","family":"Huang","sequence":"additional","affiliation":[{"name":"Departments of Data Science and AI, and Applied Mathematics, The Hong Kong Polytechnic University , Hong Kong SAR 999077, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2025,8,25]]},"reference":[{"issue":"4","key":"2026021010562066200_qkaf053-B1","first-page":"1147","article-title":"Bayesian estimation in single-index models","volume":"14","author":"Antoniadis","year":"2004","journal-title":"Statistica Sinica"},{"key":"2026021010562066200_qkaf053-B2","first-page":"214","volume-title":"Proceedings of the 34th international conference on machine learning","author":"Arjovsky","year":"2017"},{"issue":"3","key":"2026021010562066200_qkaf053-B3","doi-asserted-by":"publisher","first-page":"842","DOI":"10.1214\/13-AIHP584","article-title":"Exchangeable random measures","volume":"51","author":"Austin","year":"2015","journal-title":"Annales de l\u2019IHP Probabilit\u00e9s et Statistiques"},{"issue":"4","key":"2026021010562066200_qkaf053-B4","doi-asserted-by":"publisher","first-page":"2261","DOI":"10.1214\/18-AOS1747","article-title":"On deep learning as a remedy for the curse of dimensionality in nonparametric regression","volume":"47","author":"Bauer","year":"2019","journal-title":"Annals of Statistics"},{"key":"2026021010562066200_qkaf053-B5","volume-title":"Efficient and adaptive estimation for semiparametric models","author":"Bickel","year":"1998"},{"issue":"1","key":"2026021010562066200_qkaf053-B6","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1017\/S0266466602181096","article-title":"Regression quantiles for time series","volume":"18","author":"Cai","year":"2002","journal-title":"Econometric Theory"},{"issue":"1","key":"2026021010562066200_qkaf053-B7","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1093\/jrsssb\/qkac004","article-title":"Conformalized survival analysis","volume":"85","author":"Cand\u00e8s","year":"2023","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"issue":"1","key":"2026021010562066200_qkaf053-B8","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1111\/rssb.12476","article-title":"Inferential Wasserstein generative adversarial networks","volume":"84","author":"Chen","year":"2022","journal-title":"Journal of the Royal Statistical Society Series B"},{"key":"2026021010562066200_qkaf053-B9","first-page":"215","volume-title":"Proceedings of the 14th international conference on artificial intelligence and statistics","author":"Coates","year":"2011"},{"issue":"2","key":"2026021010562066200_qkaf053-B10","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1111\/1467-9876.00229","article-title":"Bayesian inference for generalized additive mixed models based on Markov random field priors","volume":"50","author":"Fahrmeir","year":"2001","journal-title":"Journal of the Royal Statistical Society: Series C, Applied Statistics"},{"key":"2026021010562066200_qkaf053-B11","volume-title":"Local polynomial modelling and its applications","author":"Fan","year":"1996"},{"issue":"376","key":"2026021010562066200_qkaf053-B12","doi-asserted-by":"publisher","first-page":"817","DOI":"10.1080\/01621459.1981.10477729","article-title":"Projection pursuit regression","volume":"76","author":"Friedman","year":"1981","journal-title":"Journal of the American Statistical Association"},{"key":"2026021010562066200_qkaf053-B13","first-page":"2672","volume-title":"Proceedings of the 27th international conference on neural information processing systems","author":"Goodfellow","year":"2014"},{"key":"2026021010562066200_qkaf053-B14","volume-title":"Proceedings of the 28th international conference on machine learning. Workshop on learning for global challenges","author":"Graf","year":"2011"},{"key":"2026021010562066200_qkaf053-B15","first-page":"5769","volume-title":"Proceedings of the 31st international conference on neural information processing systems","author":"Gulrajani","year":"2017"},{"key":"2026021010562066200_qkaf053-B16","doi-asserted-by":"publisher","DOI":"10.1007\/b97848","volume-title":"A distribution-free theory of nonparametric regression","author":"Gy\u00f6rfi","year":"2002"},{"issue":"463","key":"2026021010562066200_qkaf053-B17","doi-asserted-by":"publisher","first-page":"598","DOI":"10.1198\/016214503000000512","article-title":"Order-preserving nonparametric regression, with applications to conditional distribution and quantile function estimation","volume":"98","author":"Hall","year":"2003","journal-title":"Journal of the American Statistical Association"},{"issue":"445","key":"2026021010562066200_qkaf053-B18","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1080\/01621459.1999.10473832","article-title":"Methods for estimating a conditional distribution function","volume":"94","author":"Hall","year":"1999","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"2026021010562066200_qkaf053-B19","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1214\/aos\/1176349020","article-title":"Optimal smoothing in single-index models","volume":"21","author":"Hardle","year":"1993","journal-title":"Annals of Statistics"},{"issue":"3","key":"2026021010562066200_qkaf053-B20","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1214\/ss\/1177013604","article-title":"Generalized additive models","volume":"1","author":"Hastie","year":"1986","journal-title":"Statistical Science"},{"issue":"1","key":"2026021010562066200_qkaf053-B21","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1111\/rssc.12299","article-title":"Projecting UK mortality by using Bayesian generalized additive models","volume":"68","author":"Hilton","year":"2019","journal-title":"Journal of the Royal Statistical Society: Series C, Applied Statistics"},{"issue":"4","key":"2026021010562066200_qkaf053-B22","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1214\/12-STS392","article-title":"A selective review of group selection in high-dimensional models","volume":"27","author":"Huang","year":"2012","journal-title":"Statistical Science"},{"issue":"116","key":"2026021010562066200_qkaf053-B23","first-page":"1","article-title":"An error analysis of generative adversarial networks for learning distributions","volume":"23","author":"Huang","year":"2022","journal-title":"Journal of Machine Learning Research"},{"issue":"1\u20132","key":"2026021010562066200_qkaf053-B24","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/0304-4076(93)90114-K","article-title":"Semiparametric least squares (SLS) and weighted SLS estimation of single-index models","volume":"58","author":"Ichimura","year":"1993","journal-title":"Journal of Econometrics"},{"issue":"2","key":"2026021010562066200_qkaf053-B25","doi-asserted-by":"publisher","first-page":"691","DOI":"10.1214\/23-AOS2266","article-title":"Deep nonparametric regression on approximate manifolds: Nonasymptotic error bounds and polynomial prefactors","volume":"51","author":"Jiao","year":"2023","journal-title":"Annals of Statistics"},{"issue":"2","key":"2026021010562066200_qkaf053-B26","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1109\/MCI.2022.3155327","article-title":"Hands-on Bayesian neural networks\u2014a tutorial for deep learning users","volume":"17","author":"Jospin","year":"2022","journal-title":"IEEE Computational Intelligence Magazine"},{"key":"2026021010562066200_qkaf053-B27","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-4015-8","volume-title":"Foundations of modern probability","author":"Kallenberg","year":"2002"},{"issue":"509","key":"2026021010562066200_qkaf053-B28","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1080\/01621459.2014.912955","article-title":"Bayesian generalized additive models for location, scale, and shape for zero-inflated and overdispersed count data","volume":"110","author":"Klein","year":"2015","journal-title":"Journal of the American Statistical Association"},{"key":"2026021010562066200_qkaf053-B29","volume-title":"Learning multiple layers of features from tiny images","author":"Krizhevsky","year":"2009"},{"key":"2026021010562066200_qkaf053-B30","author":"LeCun","year":"2010"},{"issue":"501","key":"2026021010562066200_qkaf053-B31","doi-asserted-by":"publisher","first-page":"278","DOI":"10.1080\/01621459.2012.751873","article-title":"Distribution-free prediction sets","volume":"108","author":"Lei","year":"2013","journal-title":"Journal of the American Statistical Association"},{"issue":"3","key":"2026021010562066200_qkaf053-B32","doi-asserted-by":"publisher","first-page":"677","DOI":"10.1109\/18.382014","article-title":"Nonparametric estimation via empirical risk minimization","volume":"41","author":"Lugosi","year":"1995","journal-title":"IEEE Transactions on Information Theory"},{"issue":"7","key":"2026021010562066200_qkaf053-B33","doi-asserted-by":"publisher","first-page":"1019","DOI":"10.1016\/S0893-6080(09)80011-7","article-title":"Consistency of multilayer perceptron regression estimators","volume":"6","author":"Mielniczuk","year":"1993","journal-title":"Neural Networks"},{"issue":"2","key":"2026021010562066200_qkaf053-B34","doi-asserted-by":"publisher","first-page":"429","DOI":"10.2307\/1428011","article-title":"Integral probability metrics and their generating classes of functions","volume":"29","author":"M\u00fcller","year":"1997","journal-title":"Advances in Applied Probability"},{"issue":"1","key":"2026021010562066200_qkaf053-B35","doi-asserted-by":"publisher","first-page":"72","DOI":"10.1016\/j.jeconom.2017.06.020","article-title":"Nonparametric conditional quantile estimation: A locally weighted quantile kernel approach","volume":"201","author":"Racine","year":"2017","journal-title":"Journal of Econometrics"},{"key":"2026021010562066200_qkaf053-B36","first-page":"1060","volume-title":"Proceedings of The 33rd international conference on machine learning","author":"Reed","year":"2016"},{"issue":"3","key":"2026021010562066200_qkaf053-B37","doi-asserted-by":"publisher","first-page":"886","DOI":"10.1111\/biom.2011.67.issue-3","article-title":"Sufficient dimension reduction via Bayesian mixture modeling","volume":"67","author":"Reich","year":"2011","journal-title":"Biometrics"},{"key":"2026021010562066200_qkaf053-B38","first-page":"3543","article-title":"Conformalized quantile regression","volume":"32","author":"Romano","year":"2019","journal-title":"Advances in Neural Information Processing Systems"},{"issue":"1","key":"2026021010562066200_qkaf053-B39","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1146\/statistics.2015.2.issue-1","article-title":"Learning deep generative models","volume":"2","author":"Salakhutdinov","year":"2015","journal-title":"Annual Review of Statistics and Its Application"},{"key":"2026021010562066200_qkaf053-B40","volume-title":"Advances in neural information processing systems","author":"Salimans","year":"2016"},{"issue":"4","key":"2026021010562066200_qkaf053-B41","doi-asserted-by":"publisher","first-page":"1875","DOI":"10.1214\/19-AOS1931","article-title":"Nonparametric regression using deep neural networks with ReLU activation function","volume":"48","author":"Schmidt-Hieber","year":"2020","journal-title":"Annals of Statistics"},{"issue":"2","key":"2026021010562066200_qkaf053-B42","doi-asserted-by":"publisher","first-page":"461","DOI":"10.1214\/aos\/1176344136","article-title":"Estimating the dimension of a model","volume":"6","author":"Schwarz","year":"1978","journal-title":"Annals of Statistics"},{"key":"2026021010562066200_qkaf053-B43","doi-asserted-by":"crossref","DOI":"10.1002\/9780470316849","volume-title":"Multivariate density estimation: Theory, practice and visualization","author":"Scott","year":"1992"},{"key":"2026021010562066200_qkaf053-B44","first-page":"7694","volume-title":"International conference on artificial intelligence and statistics","author":"Sharma","year":"2023"},{"key":"2026021010562066200_qkaf053-B45","volume-title":"Density estimation for statistics and data analysis","author":"Silverman","year":"1986"},{"issue":"2","key":"2026021010562066200_qkaf053-B46","doi-asserted-by":"publisher","first-page":"590","DOI":"10.1214\/aos\/1176349940","article-title":"The dimensionality reduction principle for generalized additive models","volume":"14","author":"Stone","year":"1986","journal-title":"Annals of Statistics"},{"key":"2026021010562066200_qkaf053-B47","first-page":"261","volume-title":"International conference on indoor positioning and indoor navigation","author":"Torres-Sospedra","year":"2014"},{"key":"2026021010562066200_qkaf053-B48","volume-title":"Introduction to nonparametric estimation","author":"Tsybakov","year":"2008"},{"issue":"2","key":"2026021010562066200_qkaf053-B49","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1111\/rssb.12041","article-title":"Preadjusted non-parametric estimation of a conditional distribution function","volume":"76","author":"Veraverbeke","year":"2014","journal-title":"Journal of the Royal Statistical Society: Series B, Statistical Methodology"},{"key":"2026021010562066200_qkaf053-B50","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-71050-9","volume-title":"Optimal transport: Old and new","author":"Villani","year":"2009"},{"key":"2026021010562066200_qkaf053-B51","first-page":"444","volume-title":"International conference on machine learning","author":"Vovk","year":"1999"},{"key":"2026021010562066200_qkaf053-B52","volume-title":"All of nonparametric statistics","author":"Wasserman","year":"2006"},{"issue":"389","key":"2026021010562066200_qkaf053-B53","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1080\/01621459.1985.10477131","article-title":"Dynamic generalized linear models and Bayesian forecasting","volume":"80","author":"West","year":"1985","journal-title":"Journal of the American Statistical Association"},{"issue":"441","key":"2026021010562066200_qkaf053-B54","doi-asserted-by":"publisher","first-page":"228","DOI":"10.1080\/01621459.1998.10474104","article-title":"Local linear quantile regression","volume":"93","author":"Yu","year":"1998","journal-title":"Journal of the American Statistical Association"},{"key":"2026021010562066200_qkaf053-B55","doi-asserted-by":"publisher","first-page":"1837","DOI":"10.1080\/01621459.2021.2016424","article-title":"A deep generative approach to conditional sampling","volume":"118","author":"Zhou","year":"2022","journal-title":"Journal of the American Statistical Association"},{"key":"2026021010562066200_qkaf053-B56","first-page":"2223","volume-title":"Proceedings of the IEEE international conference on computer vision","author":"Zhu","year":"2017"}],"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\/1\/330\/64119413\/qkaf053.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/88\/1\/330\/64119413\/qkaf053.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T15:56:34Z","timestamp":1770738994000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/88\/1\/330\/8240631"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,25]]},"references-count":56,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,8,25]]},"published-print":{"date-parts":[[2026,2,10]]}},"URL":"https:\/\/doi.org\/10.1093\/jrsssb\/qkaf053","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,2]]},"published":{"date-parts":[[2025,8,25]]}}}