{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T21:31:03Z","timestamp":1782423063277,"version":"3.54.5"},"reference-count":33,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2020,9,24]],"date-time":"2020-09-24T00:00:00Z","timestamp":1600905600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,8,12]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>Wilks\u2019 theorem, which offers universal chi-squared approximations for likelihood ratio tests, is widely used in many scientific hypothesis testing problems. For modern datasets with increasing dimension, researchers have found that the conventional Wilks\u2019 phenomenon of the likelihood ratio test statistic often fails. Although new approximations have been proposed in high-dimensional settings, there still lacks a clear statistical guideline regarding how to choose between the conventional and newly proposed approximations, especially for moderate-dimensional data. To address this issue, we develop the necessary and sufficient phase transition conditions for Wilks\u2019 phenomenon under popular tests on multivariate mean and covariance structures. Moreover, we provide an in-depth analysis of the accuracy of chi-squared approximations by deriving their asymptotic biases. These results may provide helpful insights into the use of chi-squared approximations in scientific practices.<\/jats:p>","DOI":"10.1093\/biomet\/asaa078","type":"journal-article","created":{"date-parts":[[2020,9,10]],"date-time":"2020-09-10T19:12:04Z","timestamp":1599765124000},"page":"741-748","source":"Crossref","is-referenced-by-count":7,"title":["On the phase transition of Wilks\u2019 phenomenon"],"prefix":"10.1093","volume":"108","author":[{"given":"Yinqiu","family":"He","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Michigan, 323 West Hall, 1085 South University, Ann Arbor, Michigan 48109, U.S.A. yqhe@umich.edu"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bo","family":"Meng","sequence":"additional","affiliation":[{"name":"Department of Statistics and Finance, University of Science and Technology of China, Anhui, 230026, China mb0529@mail.ustc.edu.cn \u00a0zzh98052@mail.ustc.edu.cn"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhenghao","family":"Zeng","sequence":"additional","affiliation":[{"name":"Department of Statistics and Finance, University of Science and Technology of China, Anhui, 230026, China mb0529@mail.ustc.edu.cn \u00a0zzh98052@mail.ustc.edu.cn"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gongjun","family":"Xu","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Michigan, 323 West Hall, 1085 South University, Ann Arbor, Michigan 48109, U.S.A. gongjun@umich.edu"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2020,9,24]]},"reference":[{"key":"2021081218511392000_B1","article-title":"Bounds for the asymptotic distribution of the likelihood ratio","author":"Anastasiou,","year":"2018","journal-title":"arXiv"},{"key":"2021081218511392000_B2","author":"Anderson,","year":"2003","journal-title":"An Introduction to Multivariate Statistical Analysis"},{"key":"2021081218511392000_B3","doi-asserted-by":"crossref","first-page":"3822","DOI":"10.1214\/09-AOS694","article-title":"Corrections to LRT on large-dimensional covariance matrix by RMT","volume":"37","author":"Bai,","year":"2009","journal-title":"Ann. 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