{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T08:12:25Z","timestamp":1775203945135,"version":"3.50.1"},"reference-count":42,"publisher":"Oxford University Press (OUP)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2012,2,15]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Motivation: High-dimensional data such as microarrays have created new challenges to traditional statistical methods. One such example is on class prediction with high-dimension, low-sample size data. Due to the small sample size, the sample mean estimates are usually unreliable. As a consequence, the performance of the class prediction methods using the sample mean may also be unsatisfactory. To obtain more accurate estimation of parameters some statistical methods, such as regularizations through shrinkage, are often desired.<\/jats:p>\n               <jats:p>Results: In this article, we investigate the family of shrinkage estimators for the mean value under the quadratic loss function. The optimal shrinkage parameter is proposed under the scenario when the sample size is fixed and the dimension is large. We then construct a shrinkage-based diagonal discriminant rule by replacing the sample mean by the proposed shrinkage mean. Finally, we demonstrate via simulation studies and real data analysis that the proposed shrinkage-based rule outperforms its original competitor in a wide range of settings.<\/jats:p>\n               <jats:p>Contact: \u00a0tongt@hkbu.edu.hk<\/jats:p>","DOI":"10.1093\/bioinformatics\/btr690","type":"journal-article","created":{"date-parts":[[2011,12,15]],"date-time":"2011-12-15T02:00:11Z","timestamp":1323914411000},"page":"531-537","source":"Crossref","is-referenced-by-count":15,"title":["Improved mean estimation and its application to diagonal discriminant analysis"],"prefix":"10.1093","volume":"28","author":[{"given":"Tiejun","family":"Tong","sequence":"first","affiliation":[{"name":"1 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, 2Department of Biological Sciences, University of Southern California, Los Angeles, CA 90089, 3Department of Epidemiology and Public Health and 4Department of Genetics, Yale University School of Medicine, New Haven, CT 06520, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Liang","family":"Chen","sequence":"additional","affiliation":[{"name":"1 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, 2Department of Biological Sciences, University of Southern California, Los Angeles, CA 90089, 3Department of Epidemiology and Public Health and 4Department of Genetics, Yale University School of Medicine, New Haven, CT 06520, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongyu","family":"Zhao","sequence":"additional","affiliation":[{"name":"1 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, 2Department of Biological Sciences, University of Southern California, Los Angeles, CA 90089, 3Department of Epidemiology and Public Health and 4Department of Genetics, Yale University School of Medicine, New Haven, CT 06520, USA"},{"name":"1 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, 2Department of Biological Sciences, University of Southern California, Los Angeles, CA 90089, 3Department of Epidemiology and Public Health and 4Department of Genetics, Yale University School of Medicine, New Haven, CT 06520, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2011,12,14]]},"reference":[{"key":"2023012512181651600_B1","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1215\/S0012-7094-97-08808-6","article-title":"Strong laws for weighted sums of independent identically distributed random variables","volume":"88","author":"Assani","year":"1997","journal-title":"Duke Math. J."},{"key":"2023012512181651600_B2","doi-asserted-by":"crossref","first-page":"55","DOI":"10.2174\/157489306775330615","article-title":"Gene expression profile classification: a review","volume":"1","author":"Asyali","year":"2006","journal-title":"Curr. Bioinformatics"},{"key":"2023012512181651600_B3","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1093\/bioinformatics\/17.6.509","article-title":"A Bayesian framework for the analysis of microarray expression data: regularized t-test and statistical inferences of gene changes","volume":"17","author":"Baldi","year":"2001","journal-title":"Bioinformatics"},{"key":"2023012512181651600_B4","doi-asserted-by":"crossref","first-page":"642","DOI":"10.1214\/aoms\/1177697104","article-title":"A family of minimax estimators of the mean of a multivariate normal distribution","volume":"41","author":"Baranchik","year":"1970","journal-title":"Ann. Math. Stat."},{"key":"2023012512181651600_B5","doi-asserted-by":"crossref","first-page":"642","DOI":"10.1214\/aos\/1176343472","article-title":"Combining independent normal mean estimation problems with unknown variances","volume":"4","author":"Berger","year":"1976","journal-title":"Ann. Stat."},{"key":"2023012512181651600_B6","doi-asserted-by":"crossref","first-page":"989","DOI":"10.3150\/bj\/1106314847","article-title":"Some theory of Fisher's linear discriminant function, \u2018naive Bayes\u2019, and some alternatives when there are many more variables than observations","volume":"10","author":"Bickel","year":"2004","journal-title":"Bernoulli"},{"key":"2023012512181651600_B7","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/3-540-27373-5_3","article-title":"Variable selection in classification problems: a strategy based on independent component analysis","volume-title":"New Developments in Classification and Data Analysis. Studies in Classification, Data Analysis, and Knowledge Organization.","author":"Cal\u00f2","year":"2005"},{"key":"2023012512181651600_B8","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1093\/biostatistics\/kxh018","article-title":"Improved statistical tests for differential gene expression by shrinking variance components estimates","volume":"6","author":"Cui","year":"2005","journal-title":"Biostatistics"},{"key":"2023012512181651600_B9","doi-asserted-by":"crossref","first-page":"3583","DOI":"10.1093\/bioinformatics\/bth447","article-title":"Bagboosting for tumor classification with gene expression data","volume":"20","author":"Dettling","year":"2005","journal-title":"Bioinformatics"},{"key":"2023012512181651600_B10","doi-asserted-by":"crossref","first-page":"1348","DOI":"10.1093\/bioinformatics\/btg165","article-title":"Noise sampling method: an ANOVA approach allowing robust selection of differentially regulated genes measured by DNA microarrays","volume":"19","author":"Draghici","year":"2003","journal-title":"Bioinformatics"},{"key":"2023012512181651600_B11","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1198\/016214502753479248","article-title":"Comparison of discrimination methods for the classification of tumors using gene expression data","volume":"97","author":"Dudoit","year":"2002","journal-title":"J. Am. Stat. Assoc."},{"key":"2023012512181651600_B12","first-page":"117","article-title":"Stein's estimation rule and its competitors - an empirical Bayes approach","volume":"68","author":"Efron","year":"1973","journal-title":"J. Am. Stat. Assoc."},{"key":"2023012512181651600_B13","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/S0047-259X(02)00023-4","article-title":"Robust shrinkage estimation for elliptically symmetric distributions with unknown covariance matrix","volume":"85","author":"Fourdrinier","year":"2003","journal-title":"J. Multivar. Anal."},{"key":"2023012512181651600_B14","volume-title":"Bayesian Data Analysis","author":"Gelman","year":"2004","edition":"2"},{"key":"2023012512181651600_B15","doi-asserted-by":"crossref","first-page":"1625","DOI":"10.1214\/aos\/1176350184","article-title":"Minimax estimators of a normal mean vector for arbitrary quadratic loss and unknown covariance matrix","volume":"14","author":"Gleser","year":"1986","journal-title":"Ann. Stat."},{"key":"2023012512181651600_B16","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1126\/science.286.5439.531","article-title":"Molecular classification of cancer: class discovery and class prediction by gene expression monitoring","volume":"286","author":"Golub","year":"1999","journal-title":"Science"},{"key":"2023012512181651600_B17","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1093\/biostatistics\/kxj035","article-title":"Regularized linear discriminant analysis and its application in microarrays","volume":"8","author":"Guo","year":"2007","journal-title":"Biostatistics"},{"key":"2023012512181651600_B18","first-page":"1469","article-title":"Entropy inference and the James-Stein estimator, with application to nonlinear gene association networks","volume":"10","author":"Hausser","year":"2009","journal-title":"J. Mach. Learn. Res."},{"key":"2023012512181651600_B19","first-page":"84","article-title":"On the evolution of german business cycles 1958\u20132004","volume":"228","author":"Heilemann","year":"2008","journal-title":"J. Econ. Stat."},{"key":"2023012512181651600_B20","doi-asserted-by":"crossref","first-page":"1096","DOI":"10.1111\/j.1541-0420.2010.01395.x","article-title":"Bias-corrected diagonal discriminant rules for high-dimensional classification","volume":"66","author":"Huang","year":"2010","journal-title":"Biometrics"},{"key":"2023012512181651600_B21","doi-asserted-by":"crossref","first-page":"36","DOI":"10.2202\/1544-6115.1587","article-title":"Optimal tests shrinking both means and variances applicable to microarray data analysis","volume":"9","author":"Hwang","year":"2010","journal-title":"Stat. Appl. Genet. Mol. Biol."},{"key":"2023012512181651600_B22","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1111\/j.1467-9868.2008.00681.x","article-title":"Empirical Bayes confidence intervals shrinking both means and variances","volume":"71","author":"Hwang","year":"2009","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"2023012512181651600_B23","first-page":"361","article-title":"Estimation with quadratic loss","volume":"1","author":"James","year":"1961","journal-title":"Proc. Fourth Berkeley Symp. Math. Stat. Probab."},{"key":"2023012512181651600_B24","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1111\/j.1467-9868.2005.00515.x","article-title":"Estimating the proportion of true null hypotheses, with application to DNA microarray data","volume":"67","author":"Langaas","year":"2005","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"2023012512181651600_B25","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1093\/bioinformatics\/19.1.90","article-title":"Gene Selection: a Bayesian variable selection approach","volume":"19","author":"Lee","year":"2003","journal-title":"Bioinformatics"},{"key":"2023012512181651600_B26","doi-asserted-by":"crossref","first-page":"869","DOI":"10.1016\/j.csda.2004.03.017","article-title":"An extensive comparison of recent classification tools applied to microarray data","volume":"48","author":"Lee","year":"2005","journal-title":"Comput. Stat. Data Anal."},{"key":"2023012512181651600_B27","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1214\/aos\/1193342390","article-title":"Generalized Bayes minimax estimators of the multivariate normal mean with unknown covariance matrix","volume":"1","author":"Lin","year":"1973","journal-title":"Ann. Stat."},{"key":"2023012512181651600_B28","first-page":"285","article-title":"Discussion of professor Stein's paper: confidence sets for the mean of a multivariate normal distribution","volume":"24","author":"Lindley","year":"1962","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"2023012512181651600_B29","doi-asserted-by":"crossref","first-page":"23","DOI":"10.2202\/1544-6115.1504","article-title":"Shrinkage estimation of effect sizes as an alternative to hypothesis testing followed by estimation in high-dimensional biology: Applications to differential gene expression","volume":"9","author":"Montazeri","year":"2010","journal-title":"Stat. Appl. Genet. Mol. Biol."},{"key":"2023012512181651600_B30","doi-asserted-by":"crossref","first-page":"1711","DOI":"10.1016\/j.neucom.2006.01.012","article-title":"Diagonal Fisher linear discriminant analysis for efficient face recognition","volume":"69","author":"Noushath","year":"2006","journal-title":"Neurocomputing"},{"key":"2023012512181651600_B31","doi-asserted-by":"crossref","first-page":"9","DOI":"10.2202\/1544-6115.1252","article-title":"Accurate ranking of differentially expressed genes by a distribution-free shrinkage approach","volume":"6","author":"Opgen-Rhein","year":"2007","journal-title":"Stat. Appl. Genet. Mol. Biol."},{"key":"2023012512181651600_B32","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1093\/bioinformatics\/btl011","article-title":"Incorporating gene functions as priors in model-based clustering of microarray gene expression data","volume":"22","author":"Pan","year":"2006","journal-title":"Bioinformatics"},{"key":"2023012512181651600_B33","doi-asserted-by":"crossref","first-page":"1021","DOI":"10.1111\/j.1541-0420.2009.01200.x","article-title":"Shrinkage-based diagonal discriminant analysis and its applications in high-dimensional data","volume":"65","author":"Pang","year":"2009","journal-title":"Biometrics"},{"key":"2023012512181651600_B34","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1186\/1479-7364-5-1-5","article-title":"Analyzing breast cancer microarrays from african americans using shrinkage-based discriminant analysis","volume":"5","author":"Pang","year":"2010","journal-title":"Hum. Genomics"},{"key":"2023012512181651600_B35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2202\/1544-6115.1027","article-title":"Linear models and empirical Bayes methods for assessing differential expression in microarray experiment","volume":"3","author":"Smyth","year":"2004","journal-title":"Stat. Appl. Genet. Mol. Biol."},{"key":"2023012512181651600_B36","doi-asserted-by":"crossref","DOI":"10.1201\/9780203011232","volume-title":"Statistical Analysis of Gene Expression Microarray Data.","author":"Speed","year":"2003"},{"key":"2023012512181651600_B37","doi-asserted-by":"crossref","DOI":"10.1007\/0-387-21679-0_12","article-title":"SAM thresholding and false discovery rates for detecting differential gene expression in DNA microarrays","volume-title":"The Analysis of Gene Expression Data: Methods and Software.","author":"Storey","year":"2003"},{"key":"2023012512181651600_B38","doi-asserted-by":"crossref","first-page":"3170","DOI":"10.1093\/bioinformatics\/btm488","article-title":"Incorporating prior knowledge of gene functional groups into regularized discriminant analysis of microarray data","volume":"23","author":"Tai","year":"2007","journal-title":"Bioinformatics"},{"key":"2023012512181651600_B39","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1214\/ss\/1056397488","article-title":"Class prediction by nearest shrunken centroids, with applications to DNA microarrays","volume":"18","author":"Tibshirani","year":"2003","journal-title":"Stat. Sci."},{"key":"2023012512181651600_B40","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1198\/016214506000001266","article-title":"Optimal shrinkage estimation of variances with applications to microarray data analysis","volume":"102","author":"Tong","year":"2007","journal-title":"J. Am. Stat. Assoc."},{"key":"2023012512181651600_B41","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1111\/j.1467-9868.2008.00690.x","article-title":"Variance estimation in the analysis of microarray data","volume":"71","author":"Wang","year":"2009","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"2023012512181651600_B42","doi-asserted-by":"crossref","first-page":"2448","DOI":"10.1093\/bioinformatics\/btg345","article-title":"A random variance model for detection of differential gene expression in small microarray experiments","volume":"19","author":"Wright","year":"2003","journal-title":"Bioinformatics"}],"container-title":["Bioinformatics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/bioinformatics\/article-pdf\/28\/4\/531\/48873488\/bioinformatics_28_4_531.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/bioinformatics\/article-pdf\/28\/4\/531\/48873488\/bioinformatics_28_4_531.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,25]],"date-time":"2023-01-25T14:58:35Z","timestamp":1674658715000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/bioinformatics\/article\/28\/4\/531\/211887"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,12,14]]},"references-count":42,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2012,2,15]]}},"URL":"https:\/\/doi.org\/10.1093\/bioinformatics\/btr690","relation":{},"ISSN":["1367-4811","1367-4803"],"issn-type":[{"value":"1367-4811","type":"electronic"},{"value":"1367-4803","type":"print"}],"subject":[],"published-other":{"date-parts":[[2012,2,15]]},"published":{"date-parts":[[2011,12,14]]}}}