{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T18:59:50Z","timestamp":1791485990118,"version":"4.3.4"},"reference-count":55,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2025,1,17]],"date-time":"2025-01-17T00:00:00Z","timestamp":1737072000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/100000001","name":"NSF","doi-asserted-by":"publisher","award":["DMS-1844481"],"award-info":[{"award-number":["DMS-1844481"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"NSF","doi-asserted-by":"publisher","award":["DMS-2311024"],"award-info":[{"award-number":["DMS-2311024"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"NSF","doi-asserted-by":"publisher","award":["DMS-1653017"],"award-info":[{"award-number":["DMS-1653017"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,7,14]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Large datasets make it possible to build predictive models that can capture heterogenous relationships between the response variable and features. The mixture of high-dimensional linear experts model posits that observations come from a mixture of high-dimensional linear regression models, where the mixture weights are themselves feature-dependent. In this article, we show how to construct valid prediction sets for an \u21131-penalized mixture of experts model in the high-dimensional setting. We make use of a debiasing procedure to account for the bias induced by the penalization and propose a novel strategy for combining intervals to form a prediction set with coverage guarantees in the mixture setting. Synthetic examples and an application to the prediction of critical temperatures of superconducting materials show our method to have reliable practical performance.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkae117","type":"journal-article","created":{"date-parts":[[2024,12,2]],"date-time":"2024-12-02T23:22:29Z","timestamp":1733181749000},"page":"850-871","source":"Crossref","is-referenced-by-count":3,"title":["Prediction sets for high-dimensional mixture of experts models"],"prefix":"10.1093","volume":"87","author":[{"given":"Adel","family":"Javanmard","sequence":"first","affiliation":[{"name":"Data Sciences and Operations Department, Marshall School of Business, University of Southern California , 3670 Trousdale Parkway, Los Angeles, CA 90089 ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Simeng","family":"Shao","sequence":"additional","affiliation":[{"name":"Amazon , Seattle, WA ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jacob","family":"Bien","sequence":"additional","affiliation":[{"name":"Data Sciences and Operations Department, Marshall School of Business, University of Southern California , 3670 Trousdale Parkway, Los Angeles, CA 90089 ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2025,1,17]]},"reference":[{"issue":"4","key":"2026011620020978400_qkae117-B1","doi-asserted-by":"publisher","first-page":"597","DOI":"10.1111\/rssb.12268","article-title":"Approximate residual balancing: Debiased inference of average treatment effects in high dimensions","volume":"80","author":"Athey","year":"2018","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"issue":"1","key":"2026011620020978400_qkae117-B2","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1214\/16-AOS1435","article-title":"Statistical guarantees for the EM algorithm: From population to sample-based analysis","volume":"45","author":"Balakrishnan","year":"2017","journal-title":"Annals of Statistics"},{"issue":"4","key":"2026011620020978400_qkae117-B3","doi-asserted-by":"publisher","first-page":"1997","DOI":"10.1109\/TIT.2011.2174612","article-title":"The lasso risk for Gaussian matrices","volume":"58","author":"Bayati","year":"2011","journal-title":"IEEE Transactions on Information Theory"},{"key":"2026011620020978400_qkae117-B4","first-page":"28","volume-title":"Festschrift for Erich L. Lehmann. Wadsworth Statist.\/Probab. Ser","author":"Bickel","year":"1983"},{"key":"2026011620020978400_qkae117-B5","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-20192-9","volume-title":"Statistics for high-dimensional data: Methods, theory and applications","author":"B\u00fchlmann","year":"2011"},{"issue":"4","key":"2026011620020978400_qkae117-B6","doi-asserted-by":"publisher","first-page":"669","DOI":"10.1111\/rssb.12426","article-title":"Optimal statistical inference for individualized treatment effects in high-dimensional models","volume":"83","author":"Cai","year":"2021","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"issue":"2","key":"2026011620020978400_qkae117-B7","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1080\/02331888.2016.1265969","article-title":"Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity","volume":"45","author":"Cai","year":"2017","journal-title":"Annals of Statistics"},{"issue":"6","key":"2026011620020978400_qkae117-B8","doi-asserted-by":"publisher","first-page":"2313","DOI":"10.1214\/009053606000001523","article-title":"The Dantzig selector: Statistical estimation when p is much larger than n","volume":"35","author":"Candes","year":"2007","journal-title":"Annals of Statistics"},{"issue":"12","key":"2026011620020978400_qkae117-B9","doi-asserted-by":"publisher","first-page":"4497","DOI":"10.1090\/proc\/2010-138-12","article-title":"Asymptotic properties of the residual bootstrap for lasso estimators","volume":"138","author":"Chatterjee","year":"2010","journal-title":"Proceedings of the American Mathematical Society"},{"issue":"494","key":"2026011620020978400_qkae117-B10","doi-asserted-by":"publisher","first-page":"608","DOI":"10.1198\/jasa.2011.tm10159","article-title":"Bootstrapping lasso estimators","volume":"106","author":"Chatterjee","year":"2011","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"2026011620020978400_qkae117-B11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","article-title":"Maximum likelihood from incomplete data via the EM algorithm","volume":"39","author":"Dempster","year":"1977","journal-title":"Journal of the Royal Statistical Society: Series B (Methodological)"},{"key":"2026011620020978400_qkae117-B12","doi-asserted-by":"crossref","DOI":"10.1017\/9781108914062","volume-title":"Computer age statistical inference, student edition: Algorithms, evidence, and data science","author":"Efron","year":"2021"},{"key":"2026011620020978400_qkae117-B13","doi-asserted-by":"crossref","DOI":"10.1201\/9780429246593","volume-title":"An introduction to the bootstrap","author":"Efron","year":"1994"},{"issue":"5","key":"2026011620020978400_qkae117-B14","first-page":"1","article-title":"Can we trust the bootstrap in high-dimensions? The case of linear models","volume":"19","author":"El Karoui","year":"2018","journal-title":"Journal of Machine Learning Research"},{"issue":"14s","key":"2026011620020978400_qkae117-B15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/3586074","article-title":"A practical survey on faster and lighter transformers","volume":"55","author":"Fournier","year":"2023","journal-title":"ACM Computing Surveys"},{"key":"2026011620020978400_qkae117-B16","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1201\/9780429055911-12","volume-title":"Handbook of mixture analysis","author":"Gormley","year":"2019"},{"key":"2026011620020978400_qkae117-B17","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1146\/statistics.2023.10.issue-1","article-title":"Model-based clustering","volume":"10","author":"Gormley","year":"2023","journal-title":"Annual Review of Statistics and Its Application"},{"issue":"2","key":"2026011620020978400_qkae117-B18","doi-asserted-by":"publisher","first-page":"6633","DOI":"10.1214\/21-EJS1955","article-title":"Group inference in high dimensions with applications to hierarchical testing","volume":"15","author":"Guo","year":"2021","journal-title":"Electronic Journal of Statistics"},{"key":"2026011620020978400_qkae117-B19","doi-asserted-by":"publisher","first-page":"346","DOI":"10.1016\/j.commatsci.2018.07.052","article-title":"A data-driven statistical model for predicting the critical temperature of a superconductor","volume":"154","author":"Hamidieh","year":"2018","journal-title":"Computational Materials Science"},{"key":"2026011620020978400_qkae117-B20","doi-asserted-by":"publisher","author":"Hamidieh","year":"2018","DOI":"10.24432\/C53P47"},{"key":"2026011620020978400_qkae117-B21","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":"2026011620020978400_qkae117-B22","doi-asserted-by":"publisher","first-page":"623","DOI":"10.2307\/2530044","article-title":"Analysis of covariance in the mixed model: Higher-level, nonhomogeneous, and random regressions","volume":"38","author":"Henderson","year":"1982","journal-title":"Biometrics"},{"issue":"1","key":"2026011620020978400_qkae117-B23","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1198\/0003130042836","article-title":"A tutorial on mm algorithms","volume":"58","author":"Hunter","year":"2004","journal-title":"The American Statistician"},{"key":"2026011620020978400_qkae117-B24","author":"Hyun","year":"2022"},{"issue":"1","key":"2026011620020978400_qkae117-B25","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1214\/22-AOAS1631","article-title":"Modeling cell populations measured by flow cytometry with covariates using sparse mixture of regressions","volume":"17","author":"Hyun","year":"2023","journal-title":"The Annals of Applied Statistics"},{"issue":"3","key":"2026011620020978400_qkae117-B26","doi-asserted-by":"publisher","first-page":"685","DOI":"10.1111\/rssb.12373","article-title":"A flexible framework for hypothesis testing in high dimensions","volume":"82","author":"Javanmard","year":"2020","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"issue":"1","key":"2026011620020978400_qkae117-B27","first-page":"2869","article-title":"Confidence intervals and hypothesis testing for high-dimensional regression","volume":"15","author":"Javanmard","year":"2014","journal-title":"The Journal of Machine Learning Research"},{"issue":"10","key":"2026011620020978400_qkae117-B28","doi-asserted-by":"publisher","first-page":"6522","DOI":"10.1109\/TIT.2014.2343629","article-title":"Hypothesis testing in high-dimensional regression under the Gaussian random design model: Asymptotic theory","volume":"60","author":"Javanmard","year":"2014","journal-title":"IEEE Transactions on Information Theory"},{"issue":"6A","key":"2026011620020978400_qkae117-B29","doi-asserted-by":"publisher","first-page":"2593","DOI":"10.1214\/17-AOS1630","article-title":"Debiasing the lasso: Optimal sample size for gaussian designs","volume":"46","author":"Javanmard","year":"2018","journal-title":"Annals of Statistics"},{"issue":"2","key":"2026011620020978400_qkae117-B30","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1162\/neco.1994.6.2.181","article-title":"Hierarchical mixtures of experts and the em algorithm","volume":"6","author":"Jordan","year":"1994","journal-title":"Neural Computation"},{"issue":"6","key":"2026011620020978400_qkae117-B31","doi-asserted-by":"publisher","first-page":"3515","DOI":"10.1109\/TIT.18","article-title":"Estimating the coefficients of a mixture of two linear regressions by expectation maximization","volume":"65","author":"Klusowski","year":"2019","journal-title":"IEEE Transactions on Information Theory"},{"key":"2026011620020978400_qkae117-B32","author":"Kwon","year":"2020"},{"key":"2026011620020978400_qkae117-B33","author":"Kwon","year":"2019"},{"issue":"523","key":"2026011620020978400_qkae117-B34","doi-asserted-by":"publisher","first-page":"1094","DOI":"10.1080\/01621459.2017.1307116","article-title":"Distribution-free predictive inference for regression","volume":"113","author":"Lei","year":"2018","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"2026011620020978400_qkae117-B35","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1111\/rssb.12021","article-title":"Distribution-free prediction bands for non-parametric regression","volume":"76","author":"Lei","year":"2014","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"key":"2026011620020978400_qkae117-B36","author":"Papadopoulos","year":"2002"},{"key":"2026011620020978400_qkae117-B37","volume-title":"Mixed-effects models in S and S-PLUS","author":"Pinheiro","year":"2006"},{"issue":"364","key":"2026011620020978400_qkae117-B38","doi-asserted-by":"publisher","first-page":"730","DOI":"10.1080\/01621459.1978.10480085","article-title":"Estimating mixtures of normal distributions and switching regressions","volume":"73","author":"Quandt","year":"1978","journal-title":"Journal of the American Statistical Association"},{"issue":"10","key":"2026011620020978400_qkae117-B39","doi-asserted-by":"publisher","first-page":"6976","DOI":"10.1109\/TIT.2011.2165799","article-title":"Minimax rates of estimation for high-dimensional linear regression over \u2113q-balls","volume":"57","author":"Raskutti","year":"2011","journal-title":"IEEE Transactions on Information Theory"},{"issue":"2","key":"2026011620020978400_qkae117-B40","doi-asserted-by":"publisher","DOI":"10.1162\/99608f92.03f00592","article-title":"With malice towards none: Assessing uncertainty via equalized coverage","volume":"2","author":"Romano","year":"2020","journal-title":"Harvard Data Science Review"},{"key":"2026011620020978400_qkae117-B41","first-page":"3543","article-title":"Conformalized quantile regression","volume":"32","author":"Romano","year":"2019","journal-title":"Advances in Neural Information Processing Systems"},{"key":"2026011620020978400_qkae117-B42","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1007\/s11749-010-0197-z","article-title":"L1-penalization for mixture regression models","volume":"19","author":"St\u00e4dler","year":"2010","journal-title":"TEST"},{"issue":"3","key":"2026011620020978400_qkae117-B43","doi-asserted-by":"publisher","first-page":"1166","DOI":"10.1214\/14-AOS1221","article-title":"On asymptotically optimal confidence regions and tests for high-dimensional models","volume":"42","author":"Van de Geer","year":"2014","journal-title":"Annals of Statistics"},{"key":"2026011620020978400_qkae117-B44","volume-title":"Asymptotic statistics","author":"Van der Vaart","year":"2000"},{"key":"2026011620020978400_qkae117-B45","author":"Vovk","year":"2012"},{"key":"2026011620020978400_qkae117-B46","volume-title":"Algorithmic learning in a random world","author":"Vovk","year":"2005"},{"key":"2026011620020978400_qkae117-B47","first-page":"2521","article-title":"High dimensional expectation-maximization algorithm: Statistical optimization and asymptotic normality","volume":"2","author":"Wang","year":"2015","journal-title":"NIPS'15: Proceedings of the 28th International Conference on Neural Information Processing Systems"},{"key":"2026011620020978400_qkae117-B48","first-page":"1567","article-title":"Regularized EM algorithms: A unified framework and statistical guarantees","volume":"28","author":"Yi","year":"2015","journal-title":"Advances in Neural Information Processing Systems"},{"key":"2026011620020978400_qkae117-B49","author":"Yi","year":"2014"},{"issue":"8","key":"2026011620020978400_qkae117-B50","doi-asserted-by":"publisher","first-page":"1177","DOI":"10.1109\/TNNLS.2012.2200299","article-title":"Twenty years of mixture of experts","volume":"23","author":"Yuksel","year":"2012","journal-title":"IEEE transactions on neural networks and learning systems"},{"issue":"1","key":"2026011620020978400_qkae117-B51","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1111\/rssb.12026","article-title":"Confidence intervals for low dimensional parameters in high dimensional linear models","volume":"76","author":"Zhang","year":"2014","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"key":"2026011620020978400_qkae117-B52","doi-asserted-by":"publisher","author":"Zhang","year":"2020","DOI":"10.48550\/arXiv.2011.03598"},{"key":"2026011620020978400_qkae117-B53","first-page":"2541","article-title":"On model selection consistency of lasso","volume":"7","author":"Zhao","year":"2006","journal-title":"The Journal of Machine Learning Research"},{"issue":"1","key":"2026011620020978400_qkae117-B54","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1046\/j.1369-7412.2003.05379.x","article-title":"Hypothesis testing in mixture regression models","volume":"66","author":"Zhu","year":"2004","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"issue":"524","key":"2026011620020978400_qkae117-B55","doi-asserted-by":"publisher","first-page":"1583","DOI":"10.1080\/01621459.2017.1356319","article-title":"Linear hypothesis testing in dense high-dimensional linear models","volume":"113","author":"Zhu","year":"2018","journal-title":"Journal of the American Statistical Association"}],"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\/87\/3\/850\/61479177\/qkae117.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/87\/3\/850\/61479177\/qkae117.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T01:02:17Z","timestamp":1768611737000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/87\/3\/850\/7959357"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,17]]},"references-count":55,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,1,17]]},"published-print":{"date-parts":[[2025,7,14]]}},"URL":"https:\/\/doi.org\/10.1093\/jrsssb\/qkae117","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2025,7]]},"published":{"date-parts":[[2025,1,17]]}}}