{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T03:08:07Z","timestamp":1790651287874,"version":"4.1.0"},"reference-count":38,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2016,9,16]],"date-time":"2016-09-16T00:00:00Z","timestamp":1473984000000},"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\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["T32GM096982"],"award-info":[{"award-number":["T32GM096982"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000050","name":"National Heart, Lung, and Blood Institute","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000050","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007185","name":"University of California at Los Angeles","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100007185","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006196","name":"University of Oulu","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100006196","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Institute for Health and Welfare in Finland"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,9,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>Consider the following three important problems in statistical inference: constructing confidence intervals for the error of a high dimensional (p &amp;gt; n) regression estimator, the linear regression noise level and the genetic signal-to-noise ratio of a continuous-valued trait (related to the heritability). All three problems turn out to be closely related to the little-studied problem of performing inference on the l2-norm of the signal in high dimensional linear regression. We derive a novel procedure for this, which is asymptotically correct when the covariates are multivariate Gaussian and produces valid confidence intervals in finite samples as well. The procedure, called EigenPrism, is computationally fast and makes no assumptions on coefficient sparsity or knowledge of the noise level. We investigate the width of the EigenPrism confidence intervals, including a comparison with a Bayesian setting in which our interval is just 5% wider than the Bayes credible interval. We are then able to unify the three aforementioned problems by showing that EigenPrism with only minor modifications can make important contributions to all three. We also investigate the robustness of coverage and find that the method applies in practice and in finite samples much more widely than just the case of multivariate Gaussian covariates. Finally, we apply EigenPrism to a genetic data set to estimate the genetic signal-to-noise ratio for a number of continuous phenotypes.<\/jats:p>","DOI":"10.1111\/rssb.12203","type":"journal-article","created":{"date-parts":[[2016,9,17]],"date-time":"2016-09-17T03:52:56Z","timestamp":1474084376000},"page":"1037-1065","source":"Crossref","is-referenced-by-count":41,"title":["EigenPrism: Inference for High Dimensional Signal-to-Noise Ratios"],"prefix":"10.1093","volume":"79","author":[{"given":"Lucas","family":"Janson","sequence":"first","affiliation":[{"name":"Stanford University \u00a0 USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rina Foygel","family":"Barber","sequence":"additional","affiliation":[{"name":"University of Chicago \u00a0 USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Emmanuel","family":"Cand\u00e8s","sequence":"additional","affiliation":[{"name":"Stanford University \u00a0 USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2016,9,16]]},"reference":[{"key":"2023021709331774500_rssb12203-cit-0001","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1038\/nature11632","article-title":"An integrated map of genetic variation from 1, 092 human genomes","volume":"491","author":"Abecasis","year":"2012","journal-title":"Nature"},{"key":"2023021709331774500_rssb12203-cit-0002","doi-asserted-by":"crossref","first-page":"1532","DOI":"10.1214\/009117906000001079","article-title":"On asymptotics of eigenvectors of large sample covariance matrix","volume":"35","author":"Bai","year":"2007","journal-title":"Ann. 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