{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,18]],"date-time":"2026-09-18T01:49:34Z","timestamp":1789696174964,"version":"4.0.1"},"reference-count":36,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2017,9,16]],"date-time":"2017-09-16T00:00:00Z","timestamp":1505520000000},"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\/501100001809","name":"National Natural Science Fundation of China","doi-asserted-by":"publisher","award":["11401037"],"award-info":[{"award-number":["11401037"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002338","name":"Ministry of Education in China","doi-asserted-by":"publisher","award":["17YJC790057"],"award-info":[{"award-number":["17YJC790057"]}],"id":[{"id":"10.13039\/501100002338","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007931","name":"Program for Innovative Research Team of Shanghai University of Finance and Economics","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100007931","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Hong Kong Special Administrative Region Research Grants Council","award":["17332416"],"award-info":[{"award-number":["17332416"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,3,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>By studying the family of p-dimensional scale mixtures, the paper shows for the first time a non-trivial example where the eigenvalue distribution of the corresponding sample covariance matrix does not converge to the celebrated Mar\u010denko\u2013Pastur law. A different and new limit is found and characterized. The reasons for failure of the Mar\u010denko\u2013Pastur limit in this situation are found to be a strong dependence between the p-co-ordinates of the mixture. Next, we address the problem of testing whether the mixture has a spherical covariance matrix. To analyse the traditional John's-type test we establish a novel and general central limit theorem for linear statistics of eigenvalues of the sample covariance matrix. It is shown that John's test and its recent high dimensional extensions both fail for high dimensional mixtures, precisely because of the different spectral limit above. As a remedy, a new test procedure is constructed afterwards for the sphericity hypothesis. This test is then applied to identify the covariance structure in model-based clustering. It is shown that the test has much higher power than the widely used integrated classification likelihood and Bayesian information criteria in detecting non-spherical component covariance matrices of a high dimensional mixture.<\/jats:p>","DOI":"10.1111\/rssb.12248","type":"journal-article","created":{"date-parts":[[2017,9,18]],"date-time":"2017-09-18T14:00:15Z","timestamp":1505743215000},"page":"293-318","source":"Crossref","is-referenced-by-count":24,"title":["On Structure Testing for Component Covariance Matrices of a High Dimensional Mixture"],"prefix":"10.1093","volume":"80","author":[{"given":"Weiming","family":"Li","sequence":"first","affiliation":[{"name":"Shanghai University of Finance and Economics , People's Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jianfeng","family":"Yao","sequence":"additional","affiliation":[{"name":"University of Hong Kong , People's Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2017,9,16]]},"reference":[{"key":"2023022002393911900_","doi-asserted-by":"crossref","first-page":"6745","DOI":"10.1073\/pnas.96.12.6745","article-title":"Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays","volume":"96","author":"Alon","year":"1999","journal-title":"Proc. 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