{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T06:56:09Z","timestamp":1791528969936,"version":"4.3.4"},"reference-count":28,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2015,12,31]],"date-time":"2015-12-31T00:00:00Z","timestamp":1451520000000},"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":[[2017,1,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>We develop new statistical theory for probabilistic principal component analysis models in high dimensions. The focus is the estimation of the noise variance, which is an important and unresolved issue when the number of variables is large in comparison with the sample size. We first unveil the reasons for an observed downward bias of the maximum likelihood estimator of the noise variance when the data dimension is high. We then propose a bias-corrected estimator by using random-matrix theory and establish its asymptotic normality. The superiority of the new and bias-corrected estimator over existing alternatives is checked by Monte Carlo experiments with various combinations of (p, n) (the dimension and sample size). Next, we construct a new criterion based on the bias-corrected estimator to determine the number of the principal components, and a consistent estimator is obtained. Its good performance is confirmed by a simulation study and real data analysis. The bias-corrected estimator is also used to derive new asymptotics for the related goodness-of-fit statistic under the high dimensional scheme.<\/jats:p>","DOI":"10.1111\/rssb.12153","type":"journal-article","created":{"date-parts":[[2015,12,31]],"date-time":"2015-12-31T01:16:40Z","timestamp":1451524600000},"page":"51-67","source":"Crossref","is-referenced-by-count":36,"title":["On Estimation of the Noise Variance in High Dimensional Probabilistic Principal Component Analysis"],"prefix":"10.1093","volume":"79","author":[{"given":"Damien","family":"Passemier","sequence":"first","affiliation":[{"name":"University of Hong Kong \u00a0 People's Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhaoyuan","family":"Li","sequence":"additional","affiliation":[{"name":"University of Hong Kong \u00a0 People's Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jianfeng","family":"Yao","sequence":"additional","affiliation":[{"name":"University of Hong Kong \u00a0 People's Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2015,12,31]]},"reference":[{"key":"2023021709333168000_rssb12153-cit-0001","doi-asserted-by":"crossref","first-page":"1453","DOI":"10.1214\/aos\/1176347760","article-title":"Asymptotic chi-square tests for a large class of factor analysis models","volume":"18","author":"Amemiya","year":"1990","journal-title":"Ann. Statist."},{"key":"2023021709333168000_rssb12153-cit-0002","volume-title":"An Introduction to Multivariate Statistical Analysis","author":"Anderson","year":"2003"},{"key":"2023021709333168000_rssb12153-cit-0003","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1214\/aos\/1176350834","article-title":"The asymptotic normal distribution of estimators in factor analysis under general conditions","volume":"16","author":"Anderson","year":"1988","journal-title":"Ann. Statist."},{"key":"2023021709333168000_rssb12153-cit-0004","first-page":"111","volume-title":"Proc. 3rd Berkeley Symp. Mathematical Statistics and Probability","author":"Anderson","year":"1956"},{"key":"2023021709333168000_rssb12153-cit-0005","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1111\/1468-0262.00392","article-title":"Inference theory for factor models of large dimensions","volume":"71","author":"Bai","year":"2003","journal-title":"Econometrica"},{"key":"2023021709333168000_rssb12153-cit-0011","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1111\/j.1467-842X.2010.00590.x","article-title":"On estimation of the population spectral distribution from a high-dimensional sample covariance matrix","volume":"52","author":"Bai","year":"2010","journal-title":"Aust. New Zeal. J. Statist."},{"key":"2023021709333168000_rssb12153-cit-0010","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. Statist."},{"key":"2023021709333168000_rssb12153-cit-0006","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1214\/11-AOS966","article-title":"Statistical analysis of factor models of high dimension","volume":"40","author":"Bai","year":"2012","journal-title":"Ann. Statist."},{"key":"2023021709333168000_rssb12153-cit-0007","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1111\/1468-0262.00273","article-title":"Determining the number of factors in approximate factor models","volume":"70","author":"Bai","year":"2002","journal-title":"Econometrica"},{"key":"2023021709333168000_rssb12153-cit-0008","first-page":"311","article-title":"Effect of high dimension: by an example of a two sample problem","volume":"6","author":"Bai","year":"1996","journal-title":"Statist. Sin."},{"key":"2023021709333168000_rssb12153-cit-0009","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1214\/07-AIHP118","article-title":"Central limit theorems for eigenvalues in a spiked population model","volume":"44","author":"Bai","year":"2008","journal-title":"Ann. Inst. H. Poincar\u00e9 Probab. Statist."},{"key":"2023021709333168000_rssb12153-cit-0012","doi-asserted-by":"crossref","first-page":"1382","DOI":"10.1016\/j.jmva.2005.08.003","article-title":"Eigenvalues of large sample covariance matrices of spiked population models","volume":"97","author":"Baik","year":"2006","journal-title":"J. Multiv. Anal."},{"key":"2023021709333168000_rssb12153-cit-0013","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1111\/j.2044-8317.1950.tb00285.x","article-title":"Test of significance in factor analysis","volume":"3","author":"Bartlett","year":"1950","journal-title":"Br. J. Math. Statist. Psychol."},{"key":"2023021709333168000_rssb12153-cit-0014","doi-asserted-by":"crossref","first-page":"1281","DOI":"10.2307\/1912275","article-title":"Arbitrage, factor structure, and mean-variance analysis on large asset markets","volume":"51","author":"Chamberlain","year":"1983","journal-title":"Econometrica"},{"key":"2023021709333168000_rssb12153-cit-0015","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/0304-4076(95)01790-9","article-title":"Inferring the rank of a matrix","volume":"76","author":"Cragg","year":"1997","journal-title":"J. Econmetr."},{"key":"2023021709333168000_rssb12153-cit-0016","doi-asserted-by":"crossref","first-page":"3419","DOI":"10.1109\/TIP.2011.2156801","article-title":"Bayesian robust principal component analysis","volume":"20","author":"Ding","year":"2011","journal-title":"IEEE Trans. Im. Process."},{"key":"2023021709333168000_rssb12153-cit-0017","doi-asserted-by":"crossref","first-page":"1014","DOI":"10.1162\/REST_a_00225","article-title":"A quasi-maximum likelihood approach for large, approximate dynamic factor models","volume":"94","author":"Doz","year":"2012","journal-title":"Rev. Econ. Statist."},{"key":"2023021709333168000_rssb12153-cit-00500","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/j.jeconom.2008.09.017","article-title":"High dimensional covariance matrix estimation using a factor model","volume":"147","author":"Fan","year":"2008","journal-title":"J. Econmetr."},{"key":"2023021709333168000_rssb12153-cit-0018","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. B"},{"key":"2023021709333168000_rssb12153-cit-0019","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1162\/003465300559037","article-title":"The generalized dynamic factor model: identification and estimation","volume":"82","author":"Forni","year":"2000","journal-title":"Rev. Econ. Statist."},{"key":"2023021709333168000_rssb12153-cit-0020","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1214\/aos\/1009210544","article-title":"On the distribution of the largest eigenvalue in principal components analysis","volume":"29","author":"Johnstone","year":"2001","journal-title":"Ann. Statist."},{"key":"2023021709333168000_rssb12153-cit-0021","doi-asserted-by":"crossref","first-page":"682","DOI":"10.1198\/jasa.2009.0121","article-title":"On consistency and sparsity for principal components analysis in high dimensions","volume":"104","author":"Johnstone","year":"2009","journal-title":"J. Am. Statist. Ass."},{"key":"2023021709333168000_rssb12153-cit-0022","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.chemolab.2008.06.002","article-title":"Determining the number of components in a factor model from limited noisy data","volume":"94","author":"Kritchman","year":"2008","journal-title":"Chem. Int. Lab. Syst."},{"key":"2023021709333168000_rssb12153-cit-0023","doi-asserted-by":"crossref","first-page":"1447","DOI":"10.3982\/ECTA6964","article-title":"Testing hypotheses about the number of factors in large factor models","volume":"77","author":"Onatski","year":"2009","journal-title":"Econometrica"},{"key":"2023021709333168000_rssb12153-cit-0024","doi-asserted-by":"crossref","first-page":"1150002","DOI":"10.1142\/S201032631150002X","article-title":"On determining the number of spikes in a high-dimensional spiked population model","volume":"1","author":"Passemier","year":"2012","journal-title":"Rand. Matr. Theor. Appl."},{"key":"2023021709333168000_rssb12153-cit-0025","doi-asserted-by":"crossref","first-page":"5804","DOI":"10.1109\/TSP.2008.2005865","article-title":"Dimension estimation in noisy PCA with SURE and random matrix theory","volume":"56","author":"Ulfarsson","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"2023021709333168000_rssb12153-cit-0026","doi-asserted-by":"crossref","first-page":"2164","DOI":"10.1214\/13-EJS842","article-title":"On the sphericity test with large-dimensional observations","volume":"7","author":"Wang","year":"2013","journal-title":"Electron. J. Statist."},{"key":"2023021709333168000_rssb12153-cit-0027","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1214\/11-AIHP414","article-title":"Central limit theorems for linear spectral statistics of large dimensional f-matrices","volume":"48","author":"Zheng","year":"2012","journal-title":"Ann. Inst. H. Poincar\u00e9 Probab. 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.12153","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Frssb.12153","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/79\/1\/51\/49226813\/jrsssb_79_1_51.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/79\/1\/51\/49226813\/jrsssb_79_1_51.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T04:55:19Z","timestamp":1676609719000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/79\/1\/51\/7041915"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,12,31]]},"references-count":28,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,12,31]]},"published-print":{"date-parts":[[2017,1,1]]}},"URL":"https:\/\/doi.org\/10.1111\/rssb.12153","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":[[2015,12,31]]}}}