Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy-Widom limits and rates of convergence. (English) Zbl 1284.62320

Summary: Let \(A\) and \(B\) be independent, central Wishart matrices in \(p\) variables with common covariance and having \(m\) and \(n\) degrees of freedom, respectively. The distribution of the largest eigenvalue of \((A+B)^{ - 1}B\) has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that \(m\) and \(n\) grow in proportion to \(p\). We show that after centering and scaling, the distribution is approximated to second-order, \(O(p^{ - 2/3})\), by the Tracy-Widom law. The results are obtained for both complex and then real-valued data by using methods of random matrix theory to study the largest eigenvalue of the Jacobi unitary and orthogonal ensembles. Asymptotic approximations of Jacobi polynomials near the largest zero play a central role.


62H10 Multivariate distribution of statistics
62E20 Asymptotic distribution theory in statistics
60B20 Random matrices (probabilistic aspects)
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[1] Absil, P.-A., Edelman, A. and Koev, P. (2006). On the largest principal angle between random subspaces. Linear Algebra Appl. 414 288-294. · Zbl 1090.15017
[2] Adler, M., Forrester, P. J., Nagao, T. and van Moerbeke, P. (2000). Classical skew-orthogonal polynomials and random matrices. J. Statist. Phys. 99 141-170. · Zbl 0989.82020
[3] Aubin, J.-P. (1979). Applied Functional Analysis . Wiley, New York.
[4] Baik, J., Borodin, A., Deift, P. and Suidan, T. (2006). A model for the bus system in Cuernavaca (Mexico). J. Phys. A 39 8965-8975. · Zbl 1134.90438
[5] Bosbach, C. and Gawronski, W. (1999). Strong asymptotics for Jacobi polnomials with varying weights. Methods Appl. Anal. 6 39-54. · Zbl 0931.30024
[6] Carteret, H. A., Ismail, M. E. H. and Richmond, B. (2003). Three routes to the exact asymptotics for the one-dimensional quantum walk. J. Phys. A 36 8775-8795. · Zbl 1049.82025
[7] Chen, C. W. (1971). On some problems in canonical correlation analysis. Biometrika 58 399-400. JSTOR: · Zbl 0226.62058
[8] Chen, L.-C. and Ismail, M. E. H. (1991). On asymptotics of Jacobi polynomials. SIAM J. Math. Anal. 22 1442-1449. · Zbl 0735.33004
[9] Collins, B. (2005). Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Related Fields 133 315-344. · Zbl 1100.46036
[10] Constantine, A. G. (1963). Some noncentral distribution problems in multivariate analysis. Ann. Math. Statist. 34 1270-1285. · Zbl 0123.36801
[11] Deift, P. (2007). Universality for mathematical and physical systems. In Proceedings of the International Congress of Mathematicians I 125-152. Eur. Math. Soc., Zürich. · Zbl 1149.82012
[12] Deift, P. and Gioev, D. (2007). Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices. Comm. Pure Appl. Math. 60 867-910. · Zbl 1119.15022
[13] Deift, P., Gioev, D., Kriecherbauer, T. and Vanlessen, M. (2007). Universality for orthogonal and symplectic Laguerre-type ensembles. J. Statist. Phys. 129 949-1053. · Zbl 1136.15014
[14] Dumitriu, I. and Koev, P. (2008). Distributions of the extreme eigenvalues of Beta-Jacobi random matrices. SIAM J. Matrix Anal. Appl. 30 1-6. · Zbl 1158.15304
[15] Dunster, T. M. (1999). Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 281-316. · Zbl 0982.33003
[16] Dyson, F. J. (1970). Correlations between eigenvalues of a random matrix. Comm. Math. Phys. 19 235-250. · Zbl 0221.62019
[17] El Karoui, N. (2006). A rate of convergence result for the largest eigenvalue of complex white Wishart matrices. Ann. Probab. 34 2077-2117. · Zbl 1108.62014
[18] Forrester, P. J. (2004). Log-gases and Random matrices. Book manuscript. Available at http://www.ms.unimelb.edu.au/ matpjf/matpjf.html.
[19] Gohberg, I. C. and Krein, M. G. (1969). Introduction to the Theory of Linear Nonselfadjoint Operators , Translations of Mathematical Monographs 18 . Amer. Math. Soc., Providence, RI. · Zbl 0181.13504
[20] Gohberg, I., Goldberg, S. and Krupnik, N. (2000). Traces and Determinants of Linear Operators . Birkhäuser, Basel. · Zbl 0946.47013
[21] Golub, G. H. and Van Loan, C. F. (1996). Matrix Computations , 3rd ed. Johns Hopkins Univ. Press. · Zbl 0865.65009
[22] Gradshteyn, I. S. and Ryzhik, I. M. (1980). Table of Integrals, Series, and Products , corrected and enlarged edition. Translated from Russian. ed. A. Jeffrey. Academic Press, New York. · Zbl 0521.33001
[23] James, A. T. (1964). Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35 475-501. · Zbl 0121.36605
[24] Jiang, T. (2008). Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles. Probab. Theory Related Fields . DOI: 10.1007/s00440-008-0146-x. · Zbl 1162.60005
[25] Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295-327. · Zbl 1016.62078
[26] Johnstone, I. M. (2007). High-dimensional statistical inference and random matrices. In Proceedings of the International Congress of Mathematicians I 307-333. Eur. Math. Soc., Zürich. · Zbl 1120.62033
[27] Johnstone, I. M. (2009). Approximate null distribution of the largest root in multivariate analysis. Ann. Appl. Statist. · Zbl 1184.62083
[28] Khatri, C. (1972). On the exact finite series distribution of the smallest or the largest root of matrices in three situations. J. Multivariate Anal. 2 201-207. · Zbl 0271.62060
[29] Koev, P. (n.d.). Computing multivariate statistics. Manuscript in preparation.
[30] Koev, P. and Edelman, A. (2006). The efficient evaluation of the hypergeometric function of a matrix argument. Math. Comput. 75 833-846. · Zbl 1117.33007
[31] Krbalek, M. and Seba, P. (2000). The statistical properties of the city transport in Cuernavaca (Mexico) and random matrix ensembles. J. Phys. A Math. Gen. 33 L229-L234. Available at http://stacks.iop.org/0305-4470/33/L229. · Zbl 0985.90066
[32] Kuijlaars, A. B. J., McLaughlin, K. T.-R., Van Assche, W. and Vanlessen, M. (2004). The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [ - 1, 1]. Adv. in Math. 188 337-398. · Zbl 1082.42017
[33] Ma, Z. (n.d.). Accuracy of the Tracy-Widom limit for the largest eigenvalue of white Wishart matrices. Draft manuscript, Dept. Statistics, Stanford Univ. Available at arXiv.org 0810.1329. · Zbl 1248.60010
[34] Mahoux, G. and Mehta, M. L. (1991). A method of integration over matrix variables. IV. J. Phys. I (France) 1 1093-1108. · Zbl 0745.28006
[35] Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis . Academic Press, London. · Zbl 0432.62029
[36] Morrison, D. F. (2005). Multivariate Statistical Methods , 4th ed. Thomson, Belmont, CA. · Zbl 0183.20605
[37] Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory . Wiley, New York. · Zbl 0556.62028
[38] Nagao, T. and Forrester, P. J. (1995). Asymptotic correlations at the spectrum edge of random matrices. Nuclear Phys. B 435 401-420. · Zbl 1020.82588
[39] Nagao, T. and Wadati, M. (1993). Eigenvalue distribution of random matrices at the spectrum edge. J. Phys. Soc. Japan 62 3845-3856. · Zbl 0972.82543
[40] Olver, F. W. J. (1974). Asymptotics and Special Functions . Academic Press, London. · Zbl 0303.41035
[41] Riesz, F. and Sz.-Nagy, B. (1955). Functional Analysis . Ungar, New York. · Zbl 0070.10902
[42] Seiler, E. and Simon, B. (1975). An inequality for determinants. Proc. Natl. Acad. Sci. 72 3277-3288. JSTOR: · Zbl 0313.47011
[43] Simon, B. (1977). Notes on infinite determinants of Hilbert space operators. Adv. in Math. 24 244-273. · Zbl 0353.47008
[44] Szegö, G. (1967). Orthogonal Polynomials, Colloquium Publications 23 3rd ed. Amer. Math. Soc., Providence, RI. · JFM 65.0278.03
[45] Timm, N. H. (1975). Multivariate Analysis, with Applications in Education and Psychology . Brooks/Cole Publishing Cop. [Wadsworth], Monterey, CA.
[46] Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151-174. · Zbl 0789.35152
[47] Tracy, C. A. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 727-754. · Zbl 0851.60101
[48] Tracy, C. A. and Widom, H. (1998). Correlation functions, cluster functions, and spacing distributions for random matrices. J. Statist. Phys. 92 809-835. · Zbl 0942.60099
[49] Tracy, C. A. and Widom, H. (2005). Matrix kernels for the Gaussian orthogonal and symplectic ensembles. Ann. Inst. Fourier (Grenoble) 55 2197-2207. · Zbl 1084.60022
[50] Tulino, A. and Verdu, S. (2004). Random Matrix Theory and Wireless Communications . Now Publishers, Hanover, MA. · Zbl 1143.94303
[51] Wachter, K. W. (1980). The limiting empirical measure of multiple discriminant ratios. Ann. Statist. 8 937-957. · Zbl 0473.62050
[52] Widom, H. (1999). On the relation between orthogonal, symplectic and unitary ensembles. J. Statist. Phys. 94 347-363. · Zbl 0935.60090
[53] Wong, R. and Zhao, Y.-Q. (2004). Uniform asymptotic expansion of the Jacobi polynomials in a complex domain. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 2569-2586. JSTOR: · Zbl 1078.33008
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