{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,14]],"date-time":"2026-08-14T09:59:45Z","timestamp":1786701585212,"version":"build-2736575974"},"reference-count":29,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2022,11,17]],"date-time":"2022-11-17T00:00:00Z","timestamp":1668643200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,8,14]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>The Kronecker product covariance structure provides an efficient way to model the inter-correlations of matrix-variate data. In this paper, we propose test statistics for the Kronecker product covariance matrix based on linear spectral statistics of renormalized sample covariance matrices. A central limit theorem is proved for the linear spectral statistics, with explicit formulas for the mean and covariance functions, thereby filling a gap in the literature. We then show theoretically that the proposed test statistics have well-controlled size and high power. We further propose a bootstrap resampling algorithm to approximate the limiting distributions of the associated linear spectral statistics. Consistency of the bootstrap procedure is guaranteed under mild conditions. The proposed test procedure is also applicable to the Kronecker product covariance model with additional random noise. In our simulations, the empirical sizes of the proposed test procedure and its bootstrapped version are close to the corresponding theoretical values, while the power converges to $1$ quickly as the dimension and sample size increase.<\/jats:p>","DOI":"10.1093\/biomet\/asac063","type":"journal-article","created":{"date-parts":[[2022,11,16]],"date-time":"2022-11-16T13:26:27Z","timestamp":1668605187000},"page":"799-814","source":"Crossref","is-referenced-by-count":11,"title":["Testing Kronecker product covariance matrices for high-dimensional matrix-variate data"],"prefix":"10.1093","volume":"110","author":[{"given":"Long","family":"Yu","sequence":"first","affiliation":[{"name":"Shanghai University of Finance and Economics School of Statistics and Management, , 777 Guoding Road, Shanghai 200433, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiahui","family":"Xie","sequence":"additional","affiliation":[{"name":"National University of Singapore Department of Statistics and Data Science, , 21 Lower Kent Ridge Road, 117546 Singapore"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wang","family":"Zhou","sequence":"additional","affiliation":[{"name":"National University of Singapore Department of Statistics and Data Science, , 21 Lower Kent Ridge Road, 117546 Singapore"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2022,11,17]]},"reference":[{"key":"2023081420144154700_B1","doi-asserted-by":"crossref","first-page":"764","DOI":"10.1214\/09-AOAS314","article-title":"Transposable regularized covariance models with an application to missing data imputation","volume":"4","author":"Allen,","year":"2010","journal-title":"Ann. Appl. Statist."},{"key":"2023081420144154700_B2","doi-asserted-by":"crossref","first-page":"1838","DOI":"10.3150\/18-BEJ1038","article-title":"Central limit theorem for linear spectral statistics of large dimensional separable sample covariance matrices","volume":"25","author":"Bai,","year":"2019","journal-title":"Bernoulli"},{"key":"2023081420144154700_B3","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1214\/aop\/1078415845","article-title":"CLT for linear spectral statistics of large-dimensional sample covariance matrices","volume":"32","author":"Bai,","year":"2004","journal-title":"Ann. Prob."},{"key":"2023081420144154700_B4","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1214\/aop\/1176991792","article-title":"Convergence to the semicircle law","volume":"16","author":"Bai,","year":"1988","journal-title":"Ann. Prob."},{"key":"2023081420144154700_B5","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1214\/009053607000000758","article-title":"Regularized estimation of large covariance matrices","volume":"36","author":"Bickel,","year":"2008","journal-title":"Ann. Statist."},{"key":"2023081420144154700_B6","doi-asserted-by":"crossref","first-page":"672","DOI":"10.1198\/jasa.2011.tm10560","article-title":"Adaptive thresholding for sparse covariance matrix estimation","volume":"106","author":"Cai,","year":"2011","journal-title":"J. Am. Statist. Assoc."},{"key":"2023081420144154700_B7","doi-asserted-by":"crossref","first-page":"2118","DOI":"10.1214\/09-AOS752","article-title":"Optimal rates of convergence for covariance matrix estimation","volume":"38","author":"Cai,","year":"2010","journal-title":"Ann. Statist."},{"key":"2023081420144154700_B8","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.3150\/14-BEJ599","article-title":"CLT for linear spectral statistics of normalized sample covariance matrices with the dimension much larger than the sample size","volume":"21","author":"Chen,","year":"2015","journal-title":"Bernoulli"},{"key":"2023081420144154700_B9","doi-asserted-by":"crossref","first-page":"810","DOI":"10.1198\/jasa.2010.tm09560","article-title":"Tests for high-dimensional covariance matrices","volume":"105","author":"Chen,","year":"2010","journal-title":"J. Am. Statist. Assoc."},{"key":"2023081420144154700_B10","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1111\/rssb.12016","article-title":"Large covariance estimation by thresholding principal orthogonal complements","volume":"75","author":"Fan,","year":"2013","journal-title":"J. R. Statist. Soc. B"},{"key":"2023081420144154700_B11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2022.01.005","article-title":"A test for Kronecker product structure covariance matrix","author":"Guggenberger,","year":"2022","journal-title":"J. Economet. to appear"},{"key":"2023081420144154700_B12","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.spl.2016.06.027","article-title":"Testing variance parameters in models with a Kronecker product covariance structure","volume":"118","author":"Hao,","year":"2016","journal-title":"Statist. Prob. Lett."},{"key":"2023081420144154700_B13","first-page":"123","article-title":"Some optimal multivariate tests","volume":"58","author":"John,","year":"1971","journal-title":"Biometrika"},{"key":"2023081420144154700_B14","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1137\/07070111X","article-title":"Tensor decompositions and applications","volume":"51","author":"Kolda,","year":"2009","journal-title":"SIAM Rev."},{"key":"2023081420144154700_B15","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1093\/biomet\/asx014","article-title":"On the number of common factors with high-frequency data","volume":"104","author":"Kong,","year":"2017","journal-title":"Biometrika"},{"key":"2023081420144154700_B16","doi-asserted-by":"crossref","first-page":"1077","DOI":"10.1214\/17-AOS1578","article-title":"On the systematic and idiosyncratic volatility with large panel high-frequency data","volume":"46","author":"Kong,","year":"2018","journal-title":"Ann. Statist."},{"key":"2023081420144154700_B17","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1214\/aos\/1031689018","article-title":"Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size","volume":"30","author":"Ledoit,","year":"2002","journal-title":"Ann. Statist."},{"key":"2023081420144154700_B18","doi-asserted-by":"crossref","first-page":"3833","DOI":"10.3150\/17-BEJ980","article-title":"Covariance estimation via sparse Kronecker structures","volume":"24","author":"Leng,","year":"2018","journal-title":"Bernoulli"},{"key":"2023081420144154700_B19","doi-asserted-by":"crossref","first-page":"1187","DOI":"10.1080\/01621459.2012.706133","article-title":"Sparse matrix graphical models","volume":"107","author":"Leng,","year":"2012","journal-title":"J. Am. Statist. Assoc."},{"key":"2023081420144154700_B20","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.jspi.2020.06.003","article-title":"Central limit theorem for linear spectral statistics of general separable sample covariance matrices with applications","volume":"211","author":"Li,","year":"2021","journal-title":"J. Statist. Plan. Infer."},{"key":"2023081420144154700_B21","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1093\/biomet\/asz040","article-title":"Bootstrapping spectral statistics in high dimensions","volume":"106","author":"Lopes,","year":"2019","journal-title":"Biometrika"},{"key":"2023081420144154700_B22","article-title":"On likelihood-based inference for a separable covariance matrix","volume-title":"Technical report no. 337,","author":"Lu,","year":"2004"},{"key":"2023081420144154700_B23","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1016\/j.spl.2005.04.020","article-title":"The likelihood ratio test for a separable covariance matrix","volume":"73","author":"Lu,","year":"2005","journal-title":"Statist. Prob. Lett."},{"key":"2023081420144154700_B24","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1016\/j.jmva.2015.05.019","article-title":"Existence and uniqueness of the maximum likelihood estimator for models with a Kronecker product covariance structure","volume":"143","author":"Ro\u015b,","year":"2016","journal-title":"J. Mult. Anal."},{"key":"2023081420144154700_B25","doi-asserted-by":"crossref","first-page":"357","DOI":"10.3103\/S1066530708040066","article-title":"Models with a Kronecker product covariance structure: Estimation and testing","volume":"17","author":"Srivastava,","year":"2008","journal-title":"Math. Meth. Statist."},{"key":"2023081420144154700_B26","doi-asserted-by":"crossref","first-page":"3633","DOI":"10.1109\/ICASSP.2012.6288703","article-title":"Sparse covariance estimation under Kronecker product structure","volume-title":"2012 IEEE Int. Conf. Acoustics, Speech and Signal Processing (ICASSP)","author":"Tsiligkaridis,","year":"2012"},{"key":"2023081420144154700_B27","doi-asserted-by":"crossref","first-page":"5347","DOI":"10.1109\/TSP.2013.2279355","article-title":"Covariance estimation in high dimensions via Kronecker product expansions","volume":"61","author":"Tsiligkaridis,","year":"2013","journal-title":"IEEE Trans. Sig. Proces."},{"key":"2023081420144154700_B28","doi-asserted-by":"crossref","first-page":"478","DOI":"10.1109\/TSP.2007.907834","article-title":"On estimation of covariance matrices with Kronecker product structure","volume":"56","author":"Werner,","year":"2008","journal-title":"IEEE Trans. Sig. Proces."},{"key":"2023081420144154700_B29","doi-asserted-by":"crossref","first-page":"532","DOI":"10.1214\/13-AOS1187","article-title":"Gemini: Graph estimation with matrix variate normal instances","volume":"42","author":"Zhou,","year":"2014","journal-title":"Ann. Statist."}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asac063\/48936419\/asac063.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/110\/3\/799\/51111509\/asac063.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/110\/3\/799\/51111509\/asac063.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,14]],"date-time":"2023-08-14T20:35:47Z","timestamp":1692045347000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/110\/3\/799\/6832027"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,17]]},"references-count":29,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,11,17]]},"published-print":{"date-parts":[[2023,8,14]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asac063","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2023,9,1]]},"published":{"date-parts":[[2022,11,17]]}}}