Testing the independence of sets of largedimensional variables
 DanDan Jiang,
 ZhiDong Bai,
 ShuRong Zheng
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
This paper proposes the corrected likelihood ratio test (LRT) and largedimensional trace criterion to test the independence of two large sets of multivariate variables of dimensions p _{1} and p _{2} when the dimensions p = p _{1} + p _{2} and the sample size n tend to infinity simultaneously and proportionally. Both theoretical and simulation results demonstrate that the traditional χ ^{2} approximation of the LRT performs poorly when the dimension p is large relative to the sample size n, while the corrected LRT and largedimensional trace criterion behave well when the dimension is either small or large relative to the sample size. Moreover, the trace criterion can be used in the case of p > n, while the corrected LRT is unfeasible due to the loss of definition.
 Anderson T W. An Introduction to Multivariate Statistical Analysis, 3rd ed. Hoboken, NJ: John Wiley & Sons, 2003
 Bai Z D. Methodologies in spectral analysis of largedimensional random matrices: A review. Statist Sinica, 1999, 9: 611–677
 Bai Z D, Jiang D D, Yao J F, et al. Corrections to LRT on largedimensional covariance matrix by RMT. Ann Statist, 2009, 37: 3822–3840 CrossRef
 Bai Z D, Silverstein J W. CLT for linear spectral statistics of largedimensional sample covariance matrices. Ann Probab, 2004, 32: 553–605 CrossRef
 Bai Z D, Silverstein J W. Spectral Analysis of largedimensional Random Matrices, 2nd ed. Beijing: Science Press, 2010 CrossRef
 Cai T, Jiang T. Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices. Ann Statist, 2011, 39: 1496–1525 CrossRef
 Chen S X, Zhang L X, Zhong P S. Testing high dimensional covariance matrices. J Amer Statist Assoc, 2010, 105: 810–819 CrossRef
 Dempster A P. A high dimensional two sample significance test. Ann Math Statist, 1958, 29: 995–1010 CrossRef
 Fujikoshi Y, Sakurai T. Highdimensional asymptotic expansions for the distributions of canonical correlations. J Multivariate Anal, 2009, 100: 231–242 CrossRef
 Hotelling H. Relations between two sets of variants. Biometrika, 1936, 28: 321–377
 Jonsson D. Some limit theorems for the eigenvalues of a sample covariance matrix. J Multivariate Anal, 1982, 12: 1–38 CrossRef
 Ledoit O, Wolf M. Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size. Ann Statist, 2002, 30: 1081–1102 CrossRef
 Li J, Chen S X. Two sample tests for high dimensional covariance matrices. Ann Statist, 2012, 40: 908–940 CrossRef
 Schott J R. Testing for complete independence in high dimensions. Biometrika, 2005, 92: 951–956 CrossRef
 Silverstein J W. The limiting eigenvalue distribution of a multivariate F matrix. SIAM J Math Anal, 1985, 16: 641–646 CrossRef
 Sugiura N, Fujikoshi Y. Asymptotic expansions of the nonnull eistributions of the likelihood ratio criteria for multivariate linear hypothesis and independence. Ann Math Statist, 1969, 40: 942–952 CrossRef
 Wilks S S. On the independence of k sets of normally distributed statistical variables. Econometrica, 1935, 3: 309–326 CrossRef
 Yin Y Q, Bai Z D, Krishnaiah P R. Limiting behavior of the eigenvalues of a multivariate Fmatrix. J Multivariate Anal, 1983, 13: 508–516 CrossRef
 Zheng S R. Central limit theorems for linear spectral statistics of largedimensional Fmatrices. Ann Inst Henri Poincaré Probab Statist, 2012, 48: 444–476 CrossRef
 Title
 Testing the independence of sets of largedimensional variables
 Journal

Science China Mathematics
Volume 56, Issue 1 , pp 135147
 Cover Date
 20130101
 DOI
 10.1007/s1142501245010
 Print ISSN
 16747283
 Online ISSN
 18691862
 Publisher
 SP Science China Press
 Additional Links
 Topics
 Keywords

 largedimensional data analysis
 independence test
 random Fmatrices
 15A52
 62H15
 60F05
 62E20
 15A18
 Authors

 DanDan Jiang ^{(1)}
 ZhiDong Bai ^{(2)}
 ShuRong Zheng ^{(2)}
 Author Affiliations

 1. School of Mathematics, Jilin University, Changchun, 130012, China
 2. KLAS and School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China