Anderson, G. W., Guionnet, A., and Zeitouni, O. 2010. An introduction to random matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge: Cambridge University Press.Google Scholar

Anderson, T. W. 2003. An introduction to multivariate statistical analysis. 3rd edn. Hoboken, NJ: John Wiley.Google Scholar

Anderson, T. W., and Amemiya, Y. 1988. The asymptotic normal distribution of estimators in factor analysis under general conditions. Ann. Statist., 16(2), 759–771.CrossRefGoogle Scholar

Anderson, T. W., and Rubin, H. 1956. Statistical inference in factor analysis. Pages 111–150 of Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. 5. Berkeley: University of California Press.Google Scholar

Arnold, L. 1967. On the asymptotic distribution of the eigenvalues of random matrices. J. Math. Anal. Appl., 20, 262–268.CrossRefGoogle Scholar

Arnold, L. 1971. On Wigner's semicircle law for the eigenvalues of random matrices. Z. Wahrsch. Verw. Gebiete, 19, 191–198.CrossRefGoogle Scholar

Bai, Z. 1985. A note on limiting distribution of the eigenvalues of a class of random matrice. J. Math. Res. Exposition, 5(2), 113–118.Google Scholar

Bai, Z. 1999. Methodologies in spectral analysis of large dimensional random matrices: A review. Stat. Sin., 9, 611–677. With comments by G. J., Rodgers and Jack W., Silverstein; and a rejoinder by the author.Google Scholar

Bai, Z. 2005. High dimensional data analysis. Cosmos, 1(1), 17–27.CrossRefGoogle Scholar

Bai, Z., and Ding, X. 2012. Estimation of spiked eigenvalues in spiked models. Random Matrices Theory Appl., 1(2), 1150011.CrossRefGoogle Scholar

Bai, Z., and Saranadasa, H. 1996. Effect of high dimension: By an example of a two sample problem. Stat. Sin., 6(2), 311–329.Google Scholar

Bai, Z., and Silverstein, J. W. 1998. No eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices. Ann. Probab., 26, 316–345.Google Scholar

Bai, Z., and Silverstein, J. W. 1999. Exact separation of eigenvalues of large dimensional sample covariance matrices. Ann. Probab., 27(3), 1536–1555.Google Scholar

Bai, Z., and Silverstein, J. W. 2004. CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab., 32, 553–605.Google Scholar

Bai, Z., and Silverstein, J. W. 2010. Spectral Analysis of Large Dimensional Random Matrices. 2nd ed. New York: Springer.CrossRefGoogle Scholar

Bai, Z., and Yao, J. 2008. Central limit theorems for eigenvalues in a spiked population model. Ann. Inst. Henri Poincare Probab. Stat., 44(3), 447–474.Google Scholar

Bai, Z., and Yao, J. 2012. On sample eigenvalues in a generalized spiked population model. J. Multivariate Anal., 106, 167–177.CrossRefGoogle Scholar

Bai, Z., and Yin, Y. Q. 1988. A convergence to the semicircle law. Ann. Probab., 16(2), 863–875.CrossRefGoogle Scholar

Bai, Z., Yin, Y. Q., and Krishnaiah, P. R. 1986. On limiting spectral distribution of product of two random matrices when the underlying distribution is isotropic. J. Multvariate Anal., 19, 189–200.Google Scholar

Bai, Z., Yin, Y. Q., and Krishnaiah, P. R. 1987. On the limiting empirical distribution function of the eigenvalues of a multivariate F-matrix. Probab. Theory Appl., 32, 490–500.Google Scholar

Bai, Z., Miao, B. Q., and Pan, G. M. 2007. On asymptotics of eigenvectors of large sample covariance matrix. Ann. Probab., 35(4), 1532–1572.CrossRefGoogle Scholar

Bai, Z., Jiang, D., Yao, J., and Zheng, S. 2009a. Corrections to LRT on large-dimensional covariance matrix by RMT. Ann. Stat., 37(6B), 3822–3840.CrossRefGoogle Scholar

Bai, Z., Liu, H., and Wong, W. 2009b. Enhancement of the applicability of Markowitz's portfolio optimization by utilizing random matrix theory. Math. Finance, 19, 639–667.CrossRefGoogle Scholar

Bai, Z., Chen, J., and Yao, J. 2010. On estimation of the population spectral distribution from a high-dimensional sample covariance matrix. Aust.N. Z.J. Stat., 52(4), 423–437.CrossRefGoogle Scholar

Bai, Z., Li, H., and Wong, W. K. 2013a. The best estimation for high-dimensional Markowitz mean-variance optimization. Tech. rept. Northeast Normal University, Changchun.Google Scholar

Bai, Z., Jiang, D., Yao, J., and Zheng, S. 2013c. Testing linear hypotheses in high-dimensional regressions. Statistics, 47(6), 1207–1223.CrossRefGoogle Scholar

Baik, J., and Silverstein, J. W. 2006. Eigenvalues of large sample covariance matrices of spiked population models. J. Multivariate Anal., 97, 1382–1408.CrossRefGoogle Scholar

Baik, J., Ben Arous, G., and Pch, S. 2005. Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab., 33(5), 1643–1697.CrossRefGoogle Scholar

Bartlett, M. S. 1934. The vector representation of a sample. Proc. Cambridge Philos. Soc., 30, 327–340.CrossRefGoogle Scholar

Bartlett, M. S. 1937. Properties of sufficiency arid statistical tests. Proc. R. Soc. London Ser. A, 160, 268282.CrossRefGoogle Scholar

Benaych-Georges, F., and Nadakuditi, R. R. 2011. The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Adv. Math., 227(1), 494–521.CrossRefGoogle Scholar

Benaych-Georges, F., Guionnet, A., and Maida, M. 2011. Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices. Electron. J. Probab., 16(60), 1621–1662.CrossRefGoogle Scholar

Benjamini, Y., and Hochberg, Y. 1995. Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. R. Stat. Soc. Ser. B, 57, 289–300.Google Scholar

Bickel, P., and Levina, E. 2004. Some theory for Fisher's linear discriminant function ‘naive Bayes’, and some alternatives when there are many more variables than observations. Bernoulli, 10(6), 989–1010.CrossRefGoogle Scholar

Billingsley, P. 1968. Convergence of probability measures. New York: John Wiley.Google Scholar

Birke, M., and Dette, H. 2005. A note on testing the covariance matrix for large dimension. Statistics and Probability Letters, 74, 281–289.CrossRefGoogle Scholar

Bouchaud, J. P., and Potters, M. 2011. The Oxford handbook of random matrix theory. Oxford: Oxford University Press.Google Scholar

Box, G. E. P. 1949. A general distribution theory for a class of likelihood criteria. Biometrika, 36, 317–346.CrossRefGoogle ScholarPubMed

Buja, A., Hastie, T., and Tibshirani, R. 1995. Penalized discriminant analysis. Ann. Statist., 23, 73–102.Google Scholar

Canner, N., Mankiw, N. G., and Weil, D.N. 1997. An asset allocation puzzle. Am.Econ. Rev., 87(1), 181–191.Google Scholar

Capitaine, M., Donati-Martin, C., and Féral, D. 2009. The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations. Ann. Probab., 37(1), 1–47.CrossRefGoogle Scholar

Chen, J., Delyon, B., and Yao, J. 2011. On a model selection problem from high-dimensional sample covariance matrices. J. Multivariate Anal., 102, 1388–1398.CrossRefGoogle Scholar

Chen, S. X., and Qin, Y.-L. 2010. A two-sample test for high-dimensional data with applications to gene-set testing. Ann. Stat., 38, 808–835.CrossRefGoogle Scholar

Chen, S. X., Zhang, L.-X., and Zhong, P.-S. 2010. Tests for high-dimensional covariance matrices. J. Am. Stat. Assoc., 105, 810–819.CrossRefGoogle Scholar

Cheng, Y. 2004. Asymptotic probabilities of misclassification of two discriminant functions in cases of high dimensional data. Stat. Probab. Lett., 67, 9–17.CrossRefGoogle Scholar

Delyon, B. 2010. Concentration inequalities for the spectral measure of random matrices. Electron. Commun. Probab., 15, 549–562.CrossRefGoogle Scholar

Dempster, A. P. 1958. A high dimensional two sample significance test. Ann. Math. Stat., 29, 995–1010.CrossRefGoogle Scholar

Dempster, A. P. 1960. A significance test for the separation of two highly multivariate small samples. Biometrics, 16, 41–50.CrossRefGoogle Scholar

El Karoui, N. 2008. Spectrum estimation for large dimensional covariance matrices using random matrix theory. Ann. Stat., 36(6), 2757–2790.Google Scholar

Fan, J., Feng, Y., and Tong, X. 2012. A road to classification in high dimensional space: The regularized optimal affine discriminant. J. R. Stat. Soc. Ser. B, 74(4), 745–771.CrossRefGoogle Scholar

Féral, D., and Péché, S. 2007. The largest eigenvalue of rank one deformation of large Wigner matrices. Comm. Math. Phys., 272(1), 185–228.CrossRefGoogle Scholar

Fisher, R. A. 1936. The use of multiple measurements in taxonomic problems. Ann. Eugenics, 7, 179–188.CrossRefGoogle Scholar

Frankfurter, G. M., Phillips, H. E., and Seagle, J. P. 1971. Portfolio selection: The effects of uncertain means, variances and covariances. J. Finan. Quant. Anal., 6, 1251–1262.CrossRefGoogle Scholar

Giroux, A. 2013. Analyse Complexe (cours et exercices corrigés). Tech. rept. Département de mathématiques et statistique, Université de Montréal, http://dms.umontreal.ca/~giroux/.Google Scholar

Gnedenko, B. V., and Kolmogorov, A. N. 1948. Limit distributions for sums of independent random variables. Cambridge, MA: Addison-Wesley. [translated from the Russian and annotated by K. L., Chung, with appendix by J. L., Doob (1954)].Google Scholar

Golub, T. R., Slonim, D. K., Tamayo, P., Huard, C., Gaasenbeek, M., Mesirov, J. P., Coller, H., Loh, M. L., Downing, J. R., Caligiuri, M. A., Bloomfield, C. D., and Lander, E. S. 1999. Molecular classification of cancer: Class discovery and class prediction by gene expression monitoring. Science, 286(5439), 531–537.CrossRefGoogle ScholarPubMed

Grenander, U. 1963. Probabilities on Algebraic Structures. New York: John Wiley.Google Scholar

Grenander, U., and Silverstein, J. 1977. Spectral analysis of networks with random topologies. SIAMJ. Appl. Math., 32, 499–519.CrossRefGoogle Scholar

Guo, Y., Hastie, T., and Tibshirani, R. 2005. Regularized discriminant analysis and its application in microar-rays. Biostatistics, 1(1), 1–18. R package downloadable at http://cran.r-project.org/web/packages/ascrda/.Google Scholar

Hastie, T., Tibshirani, R., and Friedman, J. 2009. The elements of statistical learning. 2nd ed. New York: Springer.CrossRefGoogle Scholar

Hotelling, H. 1931. The generalization of Student's ratio. Ann. Math. Stat., 2, 360–378.CrossRefGoogle Scholar

Huber, P. J. 1973. The 1972 Wald Memorial Lecture. Robust regression: asymptotics, conjectures and Monte Carlo. Ann. Stat., 35, 73–101.Google Scholar

Jiang, D., Bai, Z., and Zheng, S. 2013. Testing the independence of sets of large-dimensional variables. Sci. China Math., 56(1), 135–147.CrossRefGoogle Scholar

Jing, B. Y., Pan, G. M., Shao, Q.-M., and Zhou, W., 2010. Nonparametric estimate of spectral density functions of sample covariance matrices: A first step. Ann. Stat., 38, 3724–3750.CrossRefGoogle Scholar

John, S. 1971. Some optimal multivariate tests. Biometrika, 58, 123–127.Google Scholar

John, S. 1972. The distribution of a statistic used for testing sphericity of normal distributions. Biometrika, 59, 169–173.CrossRefGoogle Scholar

Johnstone, I. 2001. On the distribution of the largest eigenvalue in principal components analysis. Ann. Stat., 29(2), 295–327.Google Scholar

Johnstone, I. 2007. High dimensional statistical inference and random matrices. Pages 307–333 of International Congress of Mathematicians, Vol. I. Zürich: Eur. Math. Soc.Google Scholar

Johnstone, I., and Titterington, D. 2009. Statistical challenges of high-dimensional data. Philos. Trans. R. Soc. London, Ser. A, 367(1906), 4237–4253.CrossRefGoogle ScholarPubMed

Jonsson, D. 1982. Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal., 12(1), 1–38.CrossRefGoogle Scholar

Kreĭn, M. G., and Nudel′man, A. A. 1977. The Markov Moment Problem and Extremal Problems. Providence, RI: American Mathematical Society. [Ideas and problems of P. L., Čebyšev and A. A., Markov and their further development, translated from the Russian by D., Louvish, Translations of Mathematical Monographs, Vol. 50.]CrossRefGoogle Scholar

Kritchman, S., and Nadler, B. 2008. Determining the number of components in a factor model from limited noisy data. Chem. Int. Lab. Syst., 94, 19–32.CrossRefGoogle Scholar

Kritchman, S., and Nadler, B. 2009. Non-parametric detection of the number of signals: Hypothesis testing and random matrix theory. IEEE Trans. Signal Process., 57(10), 3930–3941.CrossRefGoogle Scholar

Laloux, L., Cizeau, P. P., Bouchaud, J., and Potters, M. 1999. Noise dressing of financial correlation matrices. Phys. Rev. Lett., 83, 1467–1470.CrossRefGoogle Scholar

Ledoit, O., and Wolf, M. 2002. Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size. Ann. Stat., 30, 1081–1102.Google Scholar

Li, J., and Chen, S. X. 2012. Two sample tests for high dimensional covariance matrices. Ann. Stat., 40, 908–940.CrossRefGoogle Scholar

Li, W. M., and Yao, J. 2012. A local moments estimation of the spectrum of a large dimensional covariance matrix. Tech. rept. arXiv:1302.0356.

Li, W., Chen, J., Qin, Y., Bai, Z., and Yao, J. 2013. Estimation of the population spectral distribution from a large dimensional sample covariance matrix. J. Stat. Plann. Inference, 143 (11), 1887–1897.CrossRefGoogle Scholar

Li, Z., and Yao, J. 2014. On two simple but effective procedures for high dimensional classification of general populations. Tech. rept. arXiv:1501.01763.

Lytova, A., and Pastur, L. 2009. Central limit theorem for linear eigenvalue statistics of the Wigner and the sample covariance random matrices. Ann. Probab., 37, 1778–1840.CrossRefGoogle Scholar

Marčenko, V. A., and Pastur, L. A. 1967. Distribution of eigenvalues for some sets of random matrices. Math. USSR-Sb, 1, 457–483.CrossRefGoogle Scholar

Markowitz, H. M. 1952. Portfolio selection. J. Finance, 7, 77–91.Google Scholar

Mehta, M. L. 2004. Random matrices. 3rd ed. New York: Academic Press.

Mestre, X. 2008. Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates. IEEE Trans. Inform. Theory, 54, 5113–5129.CrossRefGoogle Scholar

Michaud, R. O. 1989. The Markowitz optimization enigma: Is “optimized” optimal?Financial Anal., 45, 31-42.Google Scholar

Nadler, B. 2010. Nonparametric detection of signals by information theoretic criteria: Performance analysis and an improved estimator. IEEE Trans. Signal Process., 58 (5), 2746–2756.CrossRefGoogle Scholar

Nagao, H. 1973a. Asymptotic expansions of the distributions of Bartlett's test and sphericity test under the local alternatives. Ann. Inst. Stat. Math., 25, 407–422.CrossRefGoogle Scholar

Nagao, H. 1973b. On some test criteria for covariance matrix. Ann. Stat., 1, 700–709.CrossRefGoogle Scholar

Nica, A., and Speicher, R. 2006. Lectures on the combinatorics offree probability. New York: Cambridge University Press.CrossRefGoogle Scholar

Pafka, S., and Kondor, I. 2004. Estimated correlation matrices and portfolio optimization. Phys. A, 343, 623-634.CrossRefGoogle Scholar

Pan, G. M. 2014. Comparison between two types of large sample covariance matrices. Ann. Inst. Henri Poincaré-Probab. Statist., 50, 655–677.CrossRefGoogle Scholar

Pan, G. M., and Zhou, W. 2008. Central limit theorem for signal-to-interference ratio of reduced rank linear receiver. Ann. Appl. Probab., 18, 1232–1270.CrossRefGoogle Scholar

Passemier, D., and Yao, J. 2012. On determining the number of spikes in a high-dimensional spiked population model. Random Matrix: Theory and Applciations, 1, 1150002.Google Scholar

Passemier, D., and Yao, J. 2013. Variance estimation and goodness-of-fit test in a high-dimensional strict factor model. Tech. rept. arXiv:1308.3890.

Passemier, D., and Yao, J. 2014. On the detection of the number of spikes, possibly equal, in the high-dimensional case. J. Multivariate Anal., 127, 173–183.CrossRefGoogle Scholar

Pastur, L. A. 1972. On the spectrum of random matrices. Theoret. Math. Phys., 10, 67–74.CrossRefGoogle Scholar

Pastur, L. A. 1973. Spectra of random self-adjoint operators. Russian Math. Surv., 28, 1–67.CrossRefGoogle Scholar

Pastur, L., and Shcherbina, M. 2011. Eigenvalue distribution of large random matrices. Mathematical Surveys and Monographs, vol. 171. Providence, RI: American Mathematical Society.Google Scholar

Paul, D. 2007. Asymptotics of sample eigenstruture for a large dimensional spiked covariance mode. Stat. Sin., 17, 1617–1642.Google Scholar

Paul, D., and Aue, A. 2014. Random matrix theory in statistics: A review. J. Stat. Plann. Inference, 150, 1-29.CrossRefGoogle Scholar

Péché, S. 2006. The largest eigenvalue of small rank perturbations of Hermitian random matrices. Probab. Theory Related Fields, 134 (1), 127–173.CrossRefGoogle Scholar

Petrov, V. V. 1975. Sums ofindependent random variables. New York: Springer.

Pizzo, A., Renfrew, D., and Soshnikov, A. 2013. On finite rank deformations of Wigner matrices. Ann. Inst. Henri Poincaré Probab. Stat., 49 (1), 64–94.CrossRefGoogle Scholar

Raj Rao, N. 2006. RMTool-A random matrix calculator in MATLAB. http://www.eecs.umich.edu/ ajnrao/rmtool/.

Rao, N. R., Mingo, J. A., Speicher, R., and Edelman, A. 2008. Statistical eigen-inference from large Wishart matrices. Ann. Statist., 36 (6), 2850–2885.CrossRefGoogle Scholar

Renfrew, D., and Soshnikov, A. 2013. On finite rank deformations of Wigner matrices II: Delocalized perturbations. Random Matrices Theory Appl., 2(1), 1250015.CrossRefGoogle Scholar

Saranadasa, H. 1993. Asymptotic expansion of the misclassification probabilities of D-and A-criteria for discrimination from two high dimensional populations using the theory of large dimensional random matrices. J. Multivariate Anal., 46, 154–174.CrossRefGoogle Scholar

Sheather, S. J., and Jones, M. C. 1991. A reliable data-based bandwidth selection method for kernel density estimation. J. R. Stat. Soc. Ser. B, 53, 683–690.Google Scholar

Silverstein, J. W. 1985. The limiting eigenvalue distribution of a multivariate F matrix. SIAM J. Math. Anal., 16(3), 641–646.CrossRefGoogle Scholar

Silverstein, J. W., and Choi, S.-I. 1995. Analysis of the limiting spectral distribution of large-dimensional random matrices. J. Multivariate Anal., 54(2), 295–309.Google Scholar

Silverstein, J. W., and Combettes, P. L. 1992. Signal detection via spectral theory of large dimensional random matrices. IEEE Trans. Signal Process., 40, 2100–2104.CrossRefGoogle Scholar

Srivastava, M. S. 2005. Some tests concerning the covariance matrix in high dimensional data. J. Jpn. Stat. Soc., 35(2), 251–272.CrossRefGoogle Scholar

Srivastava, M. S., Kollo, T., and von Rosen, D. 2011. Some tests for the covariance matrix with fewer observations than the dimension under non-normality. J. Multivariate. Anal., 102, 1090–1103.CrossRefGoogle Scholar

Sugiura, N., and Nagao, H. 1968. Unbiasedness of some test criteria for the equality of one or two covariance matrices. Ann. Math. Stat., 39, 1686–1692.CrossRefGoogle Scholar

Szegö, G. 1959. Orthogonal polynomials. New York: American Mathematical Society.Google Scholar

Tao, T. 2012. Topics in random matrix theory. Graduate Studies in Mathematics, vol. 132. Providence, RI: American Mathematical Society.CrossRefGoogle Scholar

Ulfarsson, M. O., and Solo, V. 2008. Dimension estimation in noisy PCA with SURE and random matrix theory. IEEE Trans. Signal Process., 56(12), 5804–5816.CrossRefGoogle Scholar

Wachter, K. W. 1978. The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probab., 6(1), 1–18.CrossRefGoogle Scholar

Wachter, K. W. 1980. The limiting empirical measure of multiple discriminant ratios. Ann. Stat., 8, 937–957.CrossRefGoogle Scholar

Wang, Q., and Yao, J. 2013. On the sphericity test with large-dimensional observations. Electr. J. Stat., 7, 2164–2192.Google Scholar

Wax, M., and Kailath, T. 1985. Detection of signals by information theoretic criteria. IEEE Trans. Acoust. Speech Signal Process., 33(2), 387–392.CrossRefGoogle Scholar

Wigner, E. P. 1955. Characteristic vectors bordered matrices with infinite dimensions. Ann. Math., 62, 548–564.CrossRefGoogle Scholar

Wigner, E. P. 1958. On the distributions of the roots of certain symmetric matrices. Ann. Math., 67, 325–327.CrossRefGoogle Scholar

Wilks, S. S. 1932. Certain generalizations in the analysis of variance. Biometrika, 24, 471–494.CrossRefGoogle Scholar

Wilks, S. S. 1934. Moment-generating operators for determinants of product moments in samples from a normal system. Ann. Math., 35, 312–340.CrossRefGoogle Scholar

Yin, Y. Q. 1986. Limiting spectral distribution for a class of random matrices. J. Multivariate Anal., 20(1), 50–68.CrossRefGoogle Scholar

Yin, Y. Q., and Krishnaiah, P. R. 1983. A limit theorem for the eigenvalues of product of two random matrices. J. Multivariate Anal., 13, 489–507.CrossRefGoogle Scholar

Zheng, S. 2012. Central limit theorem for linear spectral statistics of large dimensional F matrix. Ann. Inst. Henri Poincaré Probab. Statist., 48, 444–476.CrossRefGoogle Scholar

Zheng, S., Bai, Z., and Yao, J. 2015. Substitution principle for CLT of linear spectral statistics of high- dimensional sample covariance matrices with applications to hypothesis testing. The Annals ofStatistics (accepted).CrossRefGoogle Scholar

Zheng, S., Jiang, D., Bai, Z., and He, X. 2014. Inference on multiple correlation coefficients with moderately high dimensional data. Biometrika, 101(3), 748–754.CrossRefGoogle Scholar