{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,26]],"date-time":"2026-09-26T09:20:31Z","timestamp":1790414431307,"version":"4.1.0"},"reference-count":45,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2017,5,8]],"date-time":"2017-05-08T00:00:00Z","timestamp":1494201600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/501100003006","name":"Eidgen\u00f6ssiche Technische Hochschule Z\u00fcrich","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003006","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,1,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>We propose a framework for constructing goodness-of-fit tests in both low and high dimensional linear models. We advocate applying regression methods to the scaled residuals following either an ordinary least squares or lasso fit to the data, and using some proxy for prediction error as the final test statistic. We call this family residual prediction tests. We show that simulation can be used to obtain the critical values for such tests in the low dimensional setting and demonstrate using both theoretical results and extensive numerical studies that some form of the parametric bootstrap can do the same when the high dimensional linear model is under consideration. We show that residual prediction tests can be used to test for significance of groups or individual variables as special cases, and here they compare favourably with state of the art methods, but we also argue that they can be designed to test for as diverse model misspecifications as heteroscedasticity and non-linearity.<\/jats:p>","DOI":"10.1111\/rssb.12234","type":"journal-article","created":{"date-parts":[[2017,5,9]],"date-time":"2017-05-09T00:26:48Z","timestamp":1494289608000},"page":"113-135","source":"Crossref","is-referenced-by-count":38,"title":["Goodness-of-Fit Tests for High Dimensional Linear Models"],"prefix":"10.1093","volume":"80","author":[{"given":"Rajen D.","family":"Shah","sequence":"first","affiliation":[{"name":"University of Cambridge \u00a0 UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter","family":"B\u00fchlmann","sequence":"additional","affiliation":[{"name":"Eidgen\u00f6ssiche Technische Hochschule Z\u00fcrich \u00a0 Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2017,5,8]]},"reference":[{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1093\/biomet\/asr043","article-title":"Square-root lasso: pivotal recovery of sparse signals via conic programming","volume":"98","author":"Belloni","year":"2011","journal-title":"Biometrika"},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"1705","DOI":"10.1214\/08-AOS620","article-title":"Simultaneous analysis of lasso and Dantzig selector","volume":"37","author":"Bickel","year":"2009","journal-title":"Ann. Statist."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1023\/A:1010933404324","article-title":"Random forests","volume":"45","author":"Breiman","year":"2001","journal-title":"Mach. Learn."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"1212","DOI":"10.3150\/12-BEJSP11","article-title":"Statistical significance in high-dimensional linear models","volume":"19","author":"B\u00fchlmann","year":"2013","journal-title":"Bernoulli"},{"key":"2023022005320764700_","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"},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"1449","DOI":"10.1214\/15-EJS1041","article-title":"High-dimensional inference in misspecified linear models","volume":"9","author":"B\u00fchlmann","year":"2015","journal-title":"Electron. J. Statist."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1146\/annurev-statistics-022513-115545","article-title":"High-dimensional statistics with a view toward applications in biology","volume":"1","author":"B\u00fchlmann","year":"2014","journal-title":"A. Rev. Statist. Appl."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1093\/biomet\/asv039","article-title":"On the validity of the pairs bootstrap for lasso estimators","volume":"102","author":"Camponovo","year":"2015","journal-title":"Biometrika"},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"4497","DOI":"10.1090\/S0002-9939-2010-10474-4","article-title":"Asymptotic properties of the residual bootstrap for lasso estimators","volume":"138","author":"Chatterjee","year":"2010","journal-title":"Proc. Am. Math. Soc."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1198\/jasa.2011.tm10159","article-title":"Bootstrapping lasso estimators","volume":"106","author":"Chatterjee","year":"2011","journal-title":"J. Am. Statist. Ass."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511802843","volume-title":"Bootstrap Methods and Their Application","author":"Davison","year":"1997"},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1214\/15-STS527","article-title":"High-dimensional inference: confidence intervals, p-values and R-Software hdi","volume":"30","author":"Dezeure","year":"2015","journal-title":"Statist. Sci."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1214\/009053604000000067","article-title":"Least angle regression","volume":"32","author":"Efron","year":"2004","journal-title":"Ann. Statist."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","DOI":"10.1201\/9780429246593","volume-title":"An Introduction to the Bootstrap","author":"Efron","year":"1994"},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"1360","DOI":"10.1214\/09-EJS506","article-title":"On the conditions used to prove oracle results for the lasso","volume":"3","author":"van de Geer","year":"2009","journal-title":"Electron. J. Statist."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","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":"Ann. Statist."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1111\/j.1467-9868.2006.00551.x","article-title":"Testing against a high dimensional alternative","volume":"68","author":"Goeman","year":"2006","journal-title":"J. R. Statist. Soc."},{"key":"2023022005320764700_","first-page":"2869","article-title":"Confidence intervals and hypothesis testing for high-dimensional regression","volume":"15","author":"Javanmard","year":"2014","journal-title":"J. Mach. Learn. Res."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"799","DOI":"10.3150\/09-BEJ187","article-title":"The dantzig selector and sparsity oracle inequalities","volume":"15","author":"Koltchinskii","year":"2009","journal-title":"Bernoulli"},{"key":"2023022005320764700_","first-page":"18","article-title":"Classification and regression by randomforest","volume":"2","author":"Liaw","year":"2002","journal-title":"R News"},{"key":"2023022005320764700_","first-page":"413","article-title":"A significance test for the lasso","volume":"42","author":"Lockhart","year":"2014","journal-title":"Ann. Statist."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1111\/rssb.12094","article-title":"Group bound: confidence intervals for groups of variables in sparse high dimensional regression without assumptions on the design","volume":"77","author":"Meinshausen","year":"2015","journal-title":"J. R. Statist. Soc."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"1436","DOI":"10.1214\/009053606000000281","article-title":"High dimensional graphs and variable selection with the Lasso","volume":"34","author":"Meinshausen","year":"2006","journal-title":"Ann. Statist."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1111\/j.1467-9868.2010.00740.x","article-title":"Stability selection (with discussion)","volume":"72","author":"Meinshausen","year":"2010","journal-title":"J. R. Statist. Soc."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"1671","DOI":"10.1198\/jasa.2009.tm08647","article-title":"P-values for high-dimensional regression","volume":"104","author":"Meinshausen","year":"2009","journal-title":"J. Am. Statist. Ass."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1080\/10618600.2013.829780","article-title":"Variable selection diagnostics measures for high-dimensional regression","volume":"23","author":"Nan","year":"2014","journal-title":"J. Computnl Graph. Statist."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1214\/16-AOS1448","article-title":"A general theory of hypothesis tests and confidence regions for sparse high dimensional models","volume":"45","author":"Ning","year":"2017","journal-title":"Ann. Statist."},{"key":"2023022005320764700_","volume-title":"R: a Language and Environment for Statistical Computing","author":"R Development Core Team","year":"2005"},{"key":"2023022005320764700_","doi-asserted-by":"crossref","DOI":"10.5705\/ss.2014.042","article-title":"A study of error variance estimation in lasso regression","author":"Reid","year":"2016","journal-title":"Statist. Sin."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"991","DOI":"10.1214\/14-AOS1286","article-title":"Asymptotic normality and optimalities in estimation of large Gaussian graphical models","volume":"43","author":"Ren","year":"2015","journal-title":"Ann. Statist."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1111\/j.1467-9868.2011.01034.x","article-title":"Variable selection with error control: another look at stability selection","volume":"75","author":"Shah","year":"2013","journal-title":"J. R. Statist. Soc."},{"key":"2023022005320764700_","volume-title":"scalreg: scaled sparse linear regression. R Package Version 1.0","author":"Sun","year":"2013"},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1093\/biomet\/ass043","article-title":"Scaled sparse linear regression","volume":"99","author":"Sun","year":"2012","journal-title":"Biometrika"},{"key":"2023022005320764700_","first-page":"3385","article-title":"Sparse matrix inversion with scaled lasso","volume":"14","author":"Sun","year":"2013","journal-title":"J. Mach. Learn. Res."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","article-title":"Regression shrinkage and selection via the lasso","volume":"58","author":"Tibshirani","year":"1996","journal-title":"J. R. Statist. Soc."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"1456","DOI":"10.1214\/13-EJS815","article-title":"The lasso problem and uniqueness","volume":"7","author":"Tibshirani","year":"2013","journal-title":"Electron. J. Statist."},{"key":"2023022005320764700_","volume-title":"Inference in high dimensions with the penalized score test","author":"Voorman","year":"2014"},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"2178","DOI":"10.1214\/08-AOS646","article-title":"High dimensional variable selection","volume":"37","author":"Wasserman","year":"2009","journal-title":"Ann. Statist."},{"key":"2023022005320764700_","volume-title":"Resampling-based Multiple Testing: Examples and Methods for p-value Adjustment","author":"Westfall","year":"1993"},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"894","DOI":"10.1214\/09-AOS729","article-title":"Nearly unbiased variable selection under minimax concave penalty","volume":"38","author":"Zhang","year":"2010","journal-title":"Ann. Statist."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","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":"J. R. Statist. Soc."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"576","DOI":"10.1214\/12-STS399","article-title":"A general theory of concave regularization for high-dimensional sparse estimation problems","volume":"27","author":"Zhang","year":"2012","journal-title":"Statist. Sci."},{"key":"2023022005320764700_","first-page":"2541","article-title":"On model selection consistency of lasso","volume":"7","author":"Zhao","year":"2006","journal-title":"J. Mach. Learn. Res."},{"key":"2023022005320764700_","doi-asserted-by":"crossref","first-page":"1495","DOI":"10.1080\/01621459.2014.946035","article-title":"Monte Carlo simulation for lasso-type problems by estimator augmentation","volume":"109","author":"Zhou","year":"2014","journal-title":"J. Am. Statist. Ass."},{"key":"2023022005320764700_","volume-title":"Uncertainty quantification under group sparsity","author":"Zhou","year":"2015"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Frssb.12234","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/80\/1\/113\/49271429\/jrsssb_80_1_113.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/80\/1\/113\/49271429\/jrsssb_80_1_113.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,23]],"date-time":"2024-06-23T20:38:15Z","timestamp":1719175095000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/80\/1\/113\/7048421"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,5,8]]},"references-count":45,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,5,8]]},"published-print":{"date-parts":[[2018,1,1]]}},"URL":"https:\/\/doi.org\/10.1111\/rssb.12234","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2018,1]]},"published":{"date-parts":[[2017,5,8]]}}}