Abstract
This paper proposes the corrected likelihood ratio test (LRT) and large-dimensional 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 large-dimensional 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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 large-dimensional 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 large-dimensional covariance matrix by RMT. Ann Statist, 2009, 37: 3822–3840
Bai Z D, Silverstein J W. CLT for linear spectral statistics of largedimensional sample covariance matrices. Ann Probab, 2004, 32: 553–605
Bai Z D, Silverstein J W. Spectral Analysis of large-dimensional Random Matrices, 2nd ed. Beijing: Science Press, 2010
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
Chen S X, Zhang L X, Zhong P S. Testing high dimensional covariance matrices. J Amer Statist Assoc, 2010, 105: 810–819
Dempster A P. A high dimensional two sample significance test. Ann Math Statist, 1958, 29: 995–1010
Fujikoshi Y, Sakurai T. High-dimensional asymptotic expansions for the distributions of canonical correlations. J Multivariate Anal, 2009, 100: 231–242
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
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
Li J, Chen S X. Two sample tests for high dimensional covariance matrices. Ann Statist, 2012, 40: 908–940
Schott J R. Testing for complete independence in high dimensions. Biometrika, 2005, 92: 951–956
Silverstein J W. The limiting eigenvalue distribution of a multivariate F matrix. SIAM J Math Anal, 1985, 16: 641–646
Sugiura N, Fujikoshi Y. Asymptotic expansions of the non-null eistributions of the likelihood ratio criteria for multivariate linear hypothesis and independence. Ann Math Statist, 1969, 40: 942–952
Wilks S S. On the independence of k sets of normally distributed statistical variables. Econometrica, 1935, 3: 309–326
Yin Y Q, Bai Z D, Krishnaiah P R. Limiting behavior of the eigenvalues of a multivariate F-matrix. J Multivariate Anal, 1983, 13: 508–516
Zheng S R. Central limit theorems for linear spectral statistics of large-dimensional F-matrices. Ann Inst Henri Poincaré Probab Statist, 2012, 48: 444–476
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jiang, D., Bai, Z. & Zheng, S. Testing the independence of sets of large-dimensional variables. Sci. China Math. 56, 135–147 (2013). https://doi.org/10.1007/s11425-012-4501-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-012-4501-0