{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T12:53:59Z","timestamp":1791636839687,"version":"4.3.4"},"reference-count":28,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T00:00:00Z","timestamp":1676592000000},"content-version":"vor","delay-in-days":2500,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1206464"],"award-info":[{"award-number":["DMS-1206464"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1406266"],"award-info":[{"award-number":["DMS-1406266"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["R01-GM072611-9"],"award-info":[{"award-number":["R01-GM072611-9"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["R01-GM100474-4"],"award-info":[{"award-number":["R01-GM100474-4"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,1,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>Data subject to heavy-tailed errors are commonly encountered in various scientific fields. To address this problem, procedures based on quantile regression and least absolute deviation regression have been developed in recent years. These methods essentially estimate the conditional median (or quantile) function. They can be very different from the conditional mean functions, especially when distributions are asymmetric and heteroscedastic. How can we efficiently estimate the mean regression functions in ultrahigh dimensional settings with existence of only the second moment? To solve this problem, we propose a penalized Huber loss with diverging parameter to reduce biases created by the traditional Huber loss. Such a penalized robust approximate (RA) quadratic loss will be called the RA lasso. In the ultrahigh dimensional setting, where the dimensionality can grow exponentially with the sample size, our results reveal that the RA lasso estimator produces a consistent estimator at the same rate as the optimal rate under the light tail situation. We further study the computational convergence of the RA lasso and show that the composite gradient descent algorithm indeed produces a solution that admits the same optimal rate after sufficient iterations. As a by-product, we also establish the concentration inequality for estimating the population mean when there is only the second moment. We compare the RA lasso with other regularized robust estimators based on quantile regression and least absolute deviation regression. Extensive simulation studies demonstrate the satisfactory finite sample performance of the RA lasso.<\/jats:p>","DOI":"10.1111\/rssb.12166","type":"journal-article","created":{"date-parts":[[2016,4,14]],"date-time":"2016-04-14T05:16:31Z","timestamp":1460610991000},"page":"247-265","source":"Crossref","is-referenced-by-count":175,"title":["Estimation of High Dimensional Mean Regression in the Absence of Symmetry and Light Tail Assumptions"],"prefix":"10.1093","volume":"79","author":[{"given":"Jianqing","family":"Fan","sequence":"first","affiliation":[{"name":"Princeton University , USA"},{"name":"Academy of Mathematics and Systems Science , Beijing , People\u2019s Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Quefeng","family":"Li","sequence":"additional","affiliation":[{"name":"University of North Carolina at Chapel Hill \u00a0 USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuyan","family":"Wang","sequence":"additional","affiliation":[{"name":"Princeton University \u00a0 USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2016,4,14]]},"reference":[{"key":"2023021801102363800_rssb12166-cit-0001","doi-asserted-by":"crossref","first-page":"2452","DOI":"10.1214\/12-AOS1032","article-title":"Fast global convergence of gradient methods for high-dimensional statistical recovery","volume":"40","author":"Agarwal","year":"2012","journal-title":"Ann. Statist."},{"key":"2023021801102363800_rssb12166-cit-0003","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1214\/10-AOS827","article-title":"L \u00a01-penalized quantile regression in high-dimensional sparse models","volume":"39","author":"Belloni","year":"2011","journal-title":"Ann. Statist."},{"key":"2023021801102363800_rssb12166-cit-0004","doi-asserted-by":"crossref","first-page":"2577","DOI":"10.1214\/08-AOS600","article-title":"Covariance regularization by thresholding","volume":"36","author":"Bickel","year":"2008","journal-title":"Ann. Statist."},{"key":"2023021801102363800_rssb12166-cit-0005","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":"2023021801102363800_rssb12166-cit-0007","doi-asserted-by":"crossref","first-page":"1148","DOI":"10.1214\/11-AIHP454","article-title":"Challenging the empirical mean and empirical variance: a deviation study","volume":"48","author":"Catoni","year":"2012","journal-title":"Ann. Inst. H. Poincar\u00e9. Probab. Statist."},{"key":"2023021801102363800_rssb12166-cit-0008","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1145\/1390156.1390191","volume-title":"Proc. 25th Int. Conf. Machine Learning","author":"Duchi","year":"2008"},{"key":"2023021801102363800_rssb12166-cit-0009","doi-asserted-by":"crossref","first-page":"1042","DOI":"10.1198\/jasa.2010.tm09129","article-title":"Correlated z-values and the accuracy of large-scale statistical estimates","volume":"105","author":"Efron","year":"2010","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801102363800_rssb12166-cit-0010","doi-asserted-by":"crossref","first-page":"987","DOI":"10.2307\/1912773","article-title":"Autoregressive conditional heteroscedasticy with estimates of the variance of U.K. inflation","volume":"50","author":"Engle","year":"1982","journal-title":"Econometrica"},{"key":"2023021801102363800_rssb12166-cit-0011","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1214\/13-AOS1191","article-title":"Adaptive robust variable selection","volume":"42","author":"Fan","year":"2014","journal-title":"Ann. Statist."},{"key":"2023021801102363800_rssb12166-cit-0012","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1111\/j.1467-9868.2011.01005.x","article-title":"Variance estimation using refitted cross-validation in ultrahigh dimensional regression","volume":"74","author":"Fan","year":"2012","journal-title":"J. R. Statist. Soc."},{"key":"2023021801102363800_rssb12166-cit-0013","first-page":"1019","volume-title":"Estimating the false discovery proportion under arbitrary covariance dependence (with discussion)","author":"Fan","year":"2012"},{"key":"2023021801102363800_rssb12166-cit-0014","doi-asserted-by":"crossref","first-page":"1348","DOI":"10.1198\/016214501753382273","article-title":"Variable selection via nonconcave penalized likelihood and its oracle properties","volume":"96","author":"Fan","year":"2001","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801102363800_rssb12166-cit-0015","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1111\/rssb.12016","article-title":"Large covariance estimation by thresholding principal orthogonal complements (with discussion)","volume":"75","author":"Fan","year":"2013","journal-title":"J. R. Statist. Soc."},{"key":"2023021801102363800_rssb12166-cit-0016","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1111\/j.1467-9868.2008.00674.x","article-title":"Sure independence screening for ultrahigh dimensional feature space (with discussion)","volume":"70","author":"Fan","year":"2008","journal-title":"J. R. Statist. Soc."},{"key":"2023021801102363800_rssb12166-cit-0017","doi-asserted-by":"crossref","first-page":"5467","DOI":"10.1109\/TIT.2011.2158486","article-title":"Non-concave penalized likelihood with NP-Dimensionality","volume":"57","author":"Fan","year":"2011","journal-title":"IEEE Trans. Inform. Theor."},{"key":"2023021801102363800_rssb12166-cit-0018","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pone.0021067","article-title":"Activated TLR signaling in atherosclerosis among women with lower Framingham risk score: the multi-ethnic study of atherosclerosis","volume":"6","author":"Huang","year":"2011","journal-title":"PLOS ONE"},{"key":"2023021801102363800_rssb12166-cit-0019","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1214\/aoms\/1177703732","article-title":"Robust estimation of a location parameter","volume":"35","author":"Huber","year":"1964","journal-title":"Ann. Math. Statist."},{"key":"2023021801102363800_rssb12166-cit-0020","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1214\/11-EJS635","article-title":"Robust regression through the Huber's criterion and adaptive lasso penalty","volume":"5","author":"Lambert-Lacroix","year":"2011","journal-title":"Electron. J. Statist."},{"key":"2023021801102363800_rssb12166-cit-0022","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1198\/106186008X289155","article-title":"L \u00a01-norm quantile regression","volume":"17","author":"Li","year":"2008","journal-title":"J. Computnl Graph. Statist."},{"key":"2023021801102363800_rssb12166-cit-0025","doi-asserted-by":"crossref","first-page":"538","DOI":"10.1214\/12-STS400","article-title":"A unified framework for high-dimensional analysis of M-estimators with decomposable regularizers","volume":"27","author":"Negahban","year":"2012","journal-title":"Statist. Sci."},{"key":"2023021801102363800_rssb12166-cit-0026","volume-title":"Technical Report 76","author":"Nesterov","year":"2007"},{"key":"2023021801102363800_rssb12166-cit-0027","doi-asserted-by":"crossref","first-page":"6976","DOI":"10.1109\/TIT.2011.2165799","article-title":"Minimax rates of estimation for high-dimensional linear regression over L \u00a0q-balls","volume":"57","author":"Raskutti","year":"2011","journal-title":"IEEE Trans. Inform. Theor."},{"key":"2023021801102363800_rssb12166-cit-0029","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":"2023021801102363800_rssb12166-cit-0031","doi-asserted-by":"crossref","first-page":"1512","DOI":"10.1198\/jasa.2008.tm08516","article-title":"Forward regression for ultra-high dimensional variable screening","volume":"104","author":"Wang","year":"2009","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801102363800_rssb12166-cit-0032","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.jmva.2013.04.001","article-title":"The L1 penalized LAD estimator for high dimensional linear regression","volume":"120","author":"Wang","year":"2013","journal-title":"J. Multiv. Anal."},{"key":"2023021801102363800_rssb12166-cit-0033","first-page":"801","article-title":"Variable selection in quantile regression","volume":"19","author":"Wu","year":"2009","journal-title":"Statist. Sin."},{"key":"2023021801102363800_rssb12166-cit-0034","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":"2023021801102363800_rssb12166-cit-0035","doi-asserted-by":"crossref","first-page":"1108","DOI":"10.1214\/07-AOS507","article-title":"Composite quantile regression and the oracle model selection theory","volume":"36","author":"Zou","year":"2008","journal-title":"Ann. Statist."}],"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.12166","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Frssb.12166","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/rssb.12166","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1111\/rssb.12166","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/am-pdf\/10.1111\/rssb.12166","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/79\/1\/247\/49226900\/jrsssb_79_1_247.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/79\/1\/247\/49226900\/jrsssb_79_1_247.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,15]],"date-time":"2024-06-15T15:05:53Z","timestamp":1718463953000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/79\/1\/247\/7041919"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,4,14]]},"references-count":28,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,4,14]]},"published-print":{"date-parts":[[2017,1,1]]}},"URL":"https:\/\/doi.org\/10.1111\/rssb.12166","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2017,1]]},"published":{"date-parts":[[2016,4,14]]}}}