{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:23:37Z","timestamp":1740108217814,"version":"3.37.3"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,6,6]],"date-time":"2024-06-06T00:00:00Z","timestamp":1717632000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,6,6]],"date-time":"2024-06-06T00:00:00Z","timestamp":1717632000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12031016","11971324","11471223"],"award-info":[{"award-number":["12031016","11971324","11471223"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Stat"],"published-print":{"date-parts":[[2025,2]]},"DOI":"10.1007\/s00180-024-01505-1","type":"journal-article","created":{"date-parts":[[2024,6,6]],"date-time":"2024-06-06T13:01:38Z","timestamp":1717678898000},"page":"767-794","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["High dimensional T-type Estimator for robust covariance matrix estimation with applications to elliptical factor models"],"prefix":"10.1007","volume":"40","author":[{"given":"Guanpeng","family":"Wang","sequence":"first","affiliation":[]},{"given":"Hengjian","family":"Cui","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,6,6]]},"reference":[{"issue":"3","key":"1505_CR1","doi-asserted-by":"publisher","first-page":"1203","DOI":"10.3982\/ECTA8968","volume":"81","author":"S Ahn","year":"2013","unstructured":"Ahn S, Horenstein A (2013) Eigenvalue ratio test for the number of factors. Econometrica 81(3):1203\u20131227","journal-title":"Econometrica"},{"key":"1505_CR2","unstructured":"Anderson TW (2003) An introduction of multivariate statistical analysis, 3rd ed. Wiley Series in Probability and Statistics"},{"issue":"1","key":"1505_CR3","doi-asserted-by":"publisher","first-page":"436","DOI":"10.1214\/11-AOS966","volume":"40","author":"J Bai","year":"2012","unstructured":"Bai J, Li K (2012) Statistical analysis of factor models of high dimension. Ann Stat 40(1):436\u2013465","journal-title":"Ann Stat"},{"issue":"1","key":"1505_CR4","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1111\/1468-0262.00273","volume":"70","author":"J Bai","year":"2002","unstructured":"Bai J, Ng S (2002) Determining the number of factors in approximate factor models. Econometrica 70(1):191\u2013221","journal-title":"Econometrica"},{"issue":"6","key":"1505_CR5","doi-asserted-by":"publisher","first-page":"2577","DOI":"10.1214\/08-AOS600","volume":"36","author":"J Bickel","year":"2008","unstructured":"Bickel J, Levina E (2008) Covariance regularization by thresholding. Ann Stat 36(6):2577\u20132604","journal-title":"Ann Stat"},{"issue":"1","key":"1505_CR6","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1111\/j.2517-6161.1983.tb01226.x","volume":"45","author":"B Brown","year":"1983","unstructured":"Brown B (1983) Statistical uses of the spatial median. J R Stat Soc B 45(1):25\u201330","journal-title":"J R Stat Soc B"},{"key":"1505_CR7","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1198\/jasa.2011.tm10560","volume":"106","author":"T Cai","year":"2011","unstructured":"Cai T, Liu W (2011) Adaptive thresholding for sparse covariance matrix estimation. J Am Stat Assoc 106:627\u2013684","journal-title":"J Am Stat Assoc"},{"issue":"5","key":"1505_CR8","doi-asserted-by":"publisher","first-page":"2389","DOI":"10.1214\/12-AOS998","volume":"40","author":"T Cai","year":"2012","unstructured":"Cai T, Zhou H (2012) Optimal rates of convergence for sparse covariance matrix estimation. Ann Stat 40(5):2389\u20132420","journal-title":"Ann Stat"},{"key":"1505_CR9","doi-asserted-by":"publisher","first-page":"368","DOI":"10.1016\/0047-259X(81)90082-8","volume":"11","author":"S Cambanis","year":"1981","unstructured":"Cambanis S, Huang S, Simpson G (1981) On the theory of elliptically contoured distributions. J Multivar Anal 11:368\u2013385","journal-title":"J Multivar Anal"},{"issue":"9","key":"1505_CR10","doi-asserted-by":"publisher","first-page":"4097","DOI":"10.1109\/TSP.2011.2138698","volume":"59","author":"Y Chen","year":"2011","unstructured":"Chen Y, Wiesel A, Hero A (2011) Robust shrinkage estimation of high-dimensional covariance matrices. IEEE Trans Signal Process 59(9):4097\u20134107","journal-title":"IEEE Trans Signal Process"},{"issue":"2","key":"1505_CR11","doi-asserted-by":"publisher","first-page":"875","DOI":"10.3982\/ECTA15746","volume":"89","author":"L Chen","year":"2021","unstructured":"Chen L, Dolado J, Gonzalo J (2021) Quantile factor models. Econometrica 89(2):875\u2013910","journal-title":"Econometrica"},{"key":"1505_CR12","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1080\/713665670","volume":"1","author":"R Cont","year":"2001","unstructured":"Cont R (2001) Empirical properties of asset returns: stylized facts and statistical issues. Quant Finance 1:223\u2013236","journal-title":"Quant Finance"},{"issue":"4","key":"1505_CR13","doi-asserted-by":"publisher","first-page":"628","DOI":"10.1360\/03ys0020","volume":"47","author":"H Cui","year":"2004","unstructured":"Cui H (2004) On asymptotics of t-type regression estimation in multiple linear model. Sci China Ser A Math 47(4):628\u2013639","journal-title":"Sci China Ser A Math"},{"key":"1505_CR14","first-page":"5","volume":"24","author":"H Cui","year":"2005","unstructured":"Cui H (2005) $${T}$$-regression and $${L}_1$$, $${L}_2$$ approaches. J Appl Stat Manag 24:5\u201310","journal-title":"J Appl Stat Manag"},{"issue":"3","key":"1505_CR15","first-page":"321","volume":"22","author":"H Cui","year":"2006","unstructured":"Cui H (2006) T-type estimators and EM algorithm in linear model and linear errors-in-variables model. Chinese J Appl Probab Stat 22(3):321\u2013328","journal-title":"Chinese J Appl Probab Stat"},{"issue":"1","key":"1505_CR16","doi-asserted-by":"publisher","first-page":"911","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","volume":"39","author":"A Dempster","year":"1977","unstructured":"Dempster A (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc B 39(1):911\u2013924","journal-title":"J R Stat Soc B"},{"key":"1505_CR17","doi-asserted-by":"publisher","first-page":"200","DOI":"10.1016\/j.jmva.2015.11.009","volume":"144","author":"L Duembgen","year":"2016","unstructured":"Duembgen L, Nordhausen K, Schuhmacher H (2016) New algorithms for M-estimation of multivariate scatter and location. J Multivar Anal 144:200\u2013217","journal-title":"J Multivar Anal"},{"issue":"6","key":"1505_CR18","first-page":"2717","volume":"36","author":"N El Karoui","year":"2008","unstructured":"El Karoui N (2008) Operator norm consistent estimation of large dimensional sparse covariance matrices. Ann Stat 36(6):2717\u20132756","journal-title":"Ann Stat"},{"key":"1505_CR19","doi-asserted-by":"crossref","unstructured":"Fang KT, Kotz S, Ng KW (1990) Symmetric multivariate and related distributions. Monographs on Statistics and Applied Probability 36. CRC Press, London","DOI":"10.1007\/978-1-4899-2937-2"},{"key":"1505_CR20","unstructured":"Frahm G (2004) Generalized elliptical distributions: theory and applications. Ph.D. thesis, University of Koln"},{"issue":"1","key":"1505_CR21","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1214\/18-AOS1793","volume":"48","author":"J Goes","year":"2020","unstructured":"Goes J, Lerman G, Nadler B (2020) Robust sparse covariance estimation by thresholding Tyler\u2019s M-estimator. Ann Stat 48(1):86\u2013110","journal-title":"Ann Stat"},{"issue":"521","key":"1505_CR22","doi-asserted-by":"publisher","first-page":"252","DOI":"10.1080\/01621459.2016.1246366","volume":"113","author":"F Han","year":"2018","unstructured":"Han F, Liu H (2018) ECA: high-dimensional elliptical component analysis in non-Gaussian distributions. J Am Stat Assoc 113(521):252\u2013268","journal-title":"J Am Stat Assoc"},{"issue":"3","key":"1505_CR23","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1093\/biomet\/87.3.675","volume":"87","author":"X He","year":"2000","unstructured":"He X, Simpson D, Wang G (2000) Breakdown points of t-type regression estimators. Biometrika 87(3):675\u2013687","journal-title":"Biometrika"},{"issue":"1","key":"1505_CR24","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1016\/j.jspi.2003.06.002","volume":"122","author":"X He","year":"2004","unstructured":"He X, Cui H, Simpson DG (2004) Longitudinal data analysis using t-type regression. J Stat Plan Inference 122(1):253\u2013269","journal-title":"J Stat Plan Inference"},{"issue":"1","key":"1505_CR25","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1080\/07350015.2020.1811101","volume":"40","author":"Y He","year":"2022","unstructured":"He Y, Kong X, Yu L, Zhang X (2022) Large-dimensional factor analysis without moment constraints. J Bus Econ Stat 40(1):302\u2013312","journal-title":"J Bus Econ Stat"},{"issue":"4","key":"1505_CR26","doi-asserted-by":"publisher","first-page":"2102","DOI":"10.1214\/aos\/1176348388","volume":"19","author":"J Kent","year":"1991","unstructured":"Kent J, Tyler D (1991) Redescending M-estimates of multivariate location and scatter. Ann Stat 19(4):2102\u20132119","journal-title":"Ann Stat"},{"issue":"2","key":"1505_CR27","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1080\/03610919408813180","volume":"23","author":"J Kent","year":"1994","unstructured":"Kent J, Tyler D, Vard Y (1994) A curious likelihood identity for the multivariate t-distribution. Commun Stat Simul Compu 23(2):441\u2013453","journal-title":"Commun Stat Simul Compu"},{"issue":"6","key":"1505_CR28","first-page":"4254","volume":"37","author":"C Lam","year":"2009","unstructured":"Lam C, Fan J (2009) Sparsistency and rates of convergence in large covariance matrix estimation. Ann Stat 37(6):4254\u20134278","journal-title":"Ann Stat"},{"issue":"408","key":"1505_CR29","first-page":"881","volume":"84","author":"K Lange","year":"1989","unstructured":"Lange K, Little R, Taylor J (1989) Robust statistical modeling using the t distribution. J Am Stat Assoc 84(408):881\u2013896","journal-title":"J Am Stat Assoc"},{"issue":"1","key":"1505_CR30","first-page":"19","volume":"5","author":"C Liu","year":"1995","unstructured":"Liu C, Rubin DB (1995) ML estimation of the t distribution using EM and its extensions, ECM amd ECME. Stat Sin 5(1):19\u201339","journal-title":"Stat Sin"},{"key":"1505_CR31","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1016\/j.csda.2016.11.006","volume":"109","author":"R Maronna","year":"2017","unstructured":"Maronna R, Yohai V (2017) Robust and efficient estimation of multivariate scatter and location. Comput Stat Data Anal 109:64\u201375","journal-title":"Comput Stat Data Anal"},{"issue":"2","key":"1505_CR32","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1080\/10485259508832643","volume":"5","author":"J M\u00f6tt\u00f6nen","year":"1995","unstructured":"M\u00f6tt\u00f6nen J, Oja H (1995) Multivariate spatial sign and rank methods. J Nonparametric Stat 5(2):201\u2013213","journal-title":"J Nonparametric Stat"},{"key":"1505_CR33","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316559","volume-title":"Aspects of multivariate statistical theory","author":"RJ Muirhead","year":"1982","unstructured":"Muirhead RJ (1982) Aspects of multivariate statistical theory. Wiley, New York"},{"issue":"3","key":"1505_CR34","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1093\/biomet\/asv022","volume":"103","author":"K Nordhausen","year":"2015","unstructured":"Nordhausen K, Tyler D (2015) A cautionary note on robust covariance plug-in methods. Biometrika 103(3):573\u2013588","journal-title":"Biometrika"},{"issue":"4","key":"1505_CR35","doi-asserted-by":"publisher","first-page":"1004","DOI":"10.1162\/REST_a_00043","volume":"92","author":"A Onatski","year":"2010","unstructured":"Onatski A (2010) Determining the number of factors from empirical distribution of eigenvalues. Rev Econ Stat 92(4):1004\u20131016","journal-title":"Rev Econ Stat"},{"issue":"485","key":"1505_CR36","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1198\/jasa.2009.0101","volume":"104","author":"AJ Rothman","year":"2009","unstructured":"Rothman AJ, Levina E, Zhu J (2009) Generalized thresholding of largecovariance matrices. J Am Stat Assoc 104(485):177\u2013186","journal-title":"J Am Stat Assoc"},{"issue":"20","key":"1505_CR37","doi-asserted-by":"publisher","first-page":"5251","DOI":"10.1109\/TSP.2014.2348951","volume":"62","author":"I Soloveychik","year":"2014","unstructured":"Soloveychik I, Wiesel A (2014) Tyler\u2019s covariance matrix estimator in elliptical models with convex structure. IEEE Trans Signal Process 62(20):5251\u20135259","journal-title":"IEEE Trans Signal Process"},{"issue":"19","key":"1505_CR38","doi-asserted-by":"publisher","first-page":"5143","DOI":"10.1109\/TSP.2014.2348944","volume":"62","author":"Y Sun","year":"2014","unstructured":"Sun Y, Babu P, Palomar D (2014) Regularized Tyler\u2019s scatter estimator: existence, uniqueness, and algorithms. IEEE Trans Signal Process 62(19):5143\u20135156","journal-title":"IEEE Trans Signal Process"},{"issue":"14","key":"1505_CR39","doi-asserted-by":"publisher","first-page":"3576","DOI":"10.1109\/TSP.2016.2546222","volume":"64","author":"Y Sun","year":"2016","unstructured":"Sun Y, Babu P, Palomar D (2016) Robust estimation of structured covariance matrix for heavy-tailed elliptical distributions. IEEE Trans Signal Process 64(14):3576\u20133590","journal-title":"IEEE Trans Signal Process"},{"issue":"1","key":"1505_CR40","doi-asserted-by":"publisher","first-page":"234","DOI":"10.1214\/aos\/1176350263","volume":"15","author":"D Tyler","year":"1987","unstructured":"Tyler D (1987) A distribution-free M-estimator of multivariate scatter. Ann Stat 15(1):234\u2013251","journal-title":"Ann Stat"},{"key":"1505_CR41","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1016\/j.csda.2017.03.005","volume":"112","author":"Q Xia","year":"2017","unstructured":"Xia Q, Liang R, Wu J (2017) Transformed contribution ratio test for the number of factors in static approximate factor models. Comput Stat Data Anal 112:235\u2013241","journal-title":"Comput Stat Data Anal"},{"key":"1505_CR42","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1016\/j.jmva.2019.104543","volume":"174","author":"L Yu","year":"2019","unstructured":"Yu L, He Y, Zhang X (2019) Robust factor number specification for large-dimensional elliptical factor model. J Multivar Anal 174:235\u2013241","journal-title":"J Multivar Anal"}],"container-title":["Computational Statistics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-024-01505-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00180-024-01505-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-024-01505-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,17]],"date-time":"2025-02-17T21:54:54Z","timestamp":1739829294000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00180-024-01505-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,6]]},"references-count":42,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,2]]}},"alternative-id":["1505"],"URL":"https:\/\/doi.org\/10.1007\/s00180-024-01505-1","relation":{},"ISSN":["0943-4062","1613-9658"],"issn-type":[{"type":"print","value":"0943-4062"},{"type":"electronic","value":"1613-9658"}],"subject":[],"published":{"date-parts":[[2024,6,6]]},"assertion":[{"value":"25 July 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 April 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 June 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there is no conflicts related to the publication of this manuscript.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"The Ethical and informed consent for data used are not applicable for this article.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical and informed consent for data used"}}]}}