MULTIVARIATE ANALYSIS AND JACOBI ENSEMBLES: LARGEST EIGENVALUE, TRACY-WIDOM LIMITS AND RATES OF CONVERGENCE.

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.

[1]  N. L. Johnson,et al.  Multivariate Analysis , 1958, Nature.

[2]  A. Constantine.  Some Non-Central Distribution Problems in Multivariate Analysis , 1963 .

[3]  A. James.  Distributions of Matrix Variates and Latent Roots Derived from Normal Samples , 1964 .

[4]  M. Kreĭn,et al.  Introduction to the theory of linear nonselfadjoint operators , 1969 .

[5]  F. Dyson.  Correlations between eigenvalues of a random matrix , 1970 .

[6]  C. W. Chen.  On Some Problems in Canonical Correlation Analysis , 1971 .

[7]  C. Khatri.  On the exact finite series distribution of the smallest or the largest root of matrices in three situations , 1972 .

[8]  F. Olver.  Asymptotics and Special Functions , 1974 .

[9]  Barry Simon,et al.  Notes on infinite determinants of Hilbert space operators , 1977 .

[10]  G. Duncan,et al.  Multivariate Analysis: With Applications in Education and Psychology. , 1977 .

[11]  K. Wachter.  The Limiting Empirical Measure of Multiple Discriminant Ratios , 1980 .

[12]  I. S. Gradshteyn.  Table of Integrals, Series and Products, Corrected and Enlarged Edition , 1980 .

[13]  M. L. Mehta,et al.  A method of integration over matrix variables , 1981 .

[14]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[15]  M. L. Mehta,et al.  A method of integration over matrix variables: IV , 1991 .

[16]  Mourad E. H. Ismail,et al.  On asymptotics of Jacobi polynomials , 1991 .

[17]  M. Wadati,et al.  Eigenvalue Distribution of Random Matrices at the Spectrum Edge , 1993 .

[18]  C. Tracy,et al.  Level-spacing distributions and the Airy kernel , 1992, hep-th/9211141.

[19]  P. Forrester,et al.  Asymptotic correlations at the spectrum edge of random matrices , 1995 .

[20]  C. Tracy,et al.  Mathematical Physics © Springer-Verlag 1996 On Orthogonal and Symplectic Matrix Ensembles , 1995 .

[21]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[22]  Israel Gohberg,et al.  Traces and determinants of linear operators , 1996 .

[23]  Craig A. Tracy,et al.  Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices , 1998 .

[24]  Harold Widom,et al.  On the Relation Between Orthogonal, Symplectic and Unitary Matrix Ensembles , 1999 .

[25]  Christof Bosbach,et al.  Strong asymptotics for Jacobi polynomials with varying weights , 1999 .

[26]  P. Forrester,et al.  Classical Skew Orthogonal Polynomials and Random Matrices , 1999, solv-int/9907001.

[27]  M. Krbalek,et al.  The statistical properties of the city transport in Cuernavaca (Mexico) and random matrix ensembles , 2000, nlin/0001015.

[28]  I. Johnstone.  On the distribution of the largest eigenvalue in principal components analysis , 2001 .

[29]  W. Van Assche,et al.  The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1] , 2001 .

[30]  Mourad E. H. Ismail,et al.  Three routes to the exact asymptotics for the one-dimensional quantum walk , 2003, quant-ph/0303105.

[31]  Noureddine El Karoui.  A rate of convergence result for the largest eigenvalue of complex white Wishart matrices , 2004, math/0409610.

[32]  Antonia Maria Tulino,et al.  Random Matrix Theory and Wireless Communications , 2004, Found. Trends Commun. Inf. Theory.

[33]  Uniform asymptotic expansion of the Jacobi polynomials in a complex domain , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[34]  Matrix kernels for the Gaussian orthogonal and symplectic ensembles , 2004, math-ph/0405035.

[35]  Universality in Random Matrix Theory for orthogonal and symplectic ensembles , 2004, math-ph/0411075.

[36]  Benoit Collins.  Product of random projections, Jacobi ensembles and universality problems arising from free probability , 2005 .

[37]  William J. Wilson,et al.  Multivariate Statistical Methods , 2005, Technometrics.

[38]  Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices , 2005, math-ph/0507023.

[39]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[40]  L. M.,et al.  A Method of Integration over Matrix Variables , 2005 .

[41]  J. Baik,et al.  A model for the bus system in Cuernavaca (Mexico) , 2005, math/0510414.

[42]  P. Koev,et al.  On the largest principal angle between random subspaces , 2006 .

[43]  Alan Edelman,et al.  The efficient evaluation of the hypergeometric function of a matrix argument , 2006, Math. Comput..

[44]  I. Johnstone.  High Dimensional Statistical Inference and Random Matrices , 2006, math/0611589.

[45]  P. Deift.  Universality for mathematical and physical systems , 2006, math-ph/0603038.

[46]  Universality for Orthogonal and Symplectic Laguerre-Type Ensembles , 2006, math-ph/0612007.

[47]  Ioana Dumitriu,et al.  Distributions of the Extreme Eigenvaluesof Beta-Jacobi Random Matrices , 2008, SIAM J. Matrix Anal. Appl..

[48]  Zongming Ma,et al.  Accuracy of the Tracy-Widom limit for the largest eigenvalue in white Wishart matrices , 2008, 0810.1329.

[49]  Tiefeng Jiang,et al.  Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles , 2009 .

[50]  Iain M Johnstone,et al.  APPROXIMATE NULL DISTRIBUTION OF THE LARGEST ROOT IN MULTIVARIATE ANALYSIS. , 2010, The annals of applied statistics.

[51]  P. Forrester.  Log-Gases and Random Matrices , 2010 .