On the distribution of the largest eigenvalue in principal components analysis

Annals of Statistics

Volumn 29, Issue 2, 2001, Pages 295-327

  Johnstone, Iain M ^a  

^a STANFORD UNIVERSITY   (United States)

Author keywords

Empirical orthogonal functions; Karhunen Lo ve transform; Laguerre ensemble; Laguerre polynomial; Largest eigenvalue; Largest singular value; Painlev equation; Plancherel Rotach asymptotics; Tracy Widom distribution; Wishart distribution

Indexed keywords

EID: 0035641726     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1009210544     Document Type: Article

References (49)

1. ALDOUS, D. and DIACONIS, P. (1999). Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem. Bull. Amer. Math. Soc. 36 413-432.
2. ANDERSON, T. W. (1963). Asymptotic theory for principal component analysis. Ann Math. Statist. 34 122-148
3. ANDERSON, T. W. (1996). R. A. Fisher and multivariate analysis. Statist. Sci. 11 20-34.
4. BAI, Z. D. (1999). Methodologies in spectral analysis of large dimensional random matrices: a review. Statist. Sinica 9 611-677.
5. BAIK, J., DEIFT, P. and JOHANSSON, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119-1178.
6. BAKER, T. H., FORRESTER, P. J. and PEARCE, P. A. (1998). Random matrix ensembles with an effective extensive external charge. J. Phys. A 31 6087-6101.
7. BASOR, E. L. (1997). Distribution functions for random variables for ensembles of positive Hermitian matrices, Comm. Math. Phys. 188 327-350.
8. BUJA, A., HASTIE, T. and TIBSHIRANI, R. (1995). Penalized discriminant analysis. Ann. Statist. 23 73-102.
9. CONSTANTINE, A. G. (1963). Some non-central distribution problems in multivariate analysis. Ann. Math. Statist. 34 1270-1285.
10. DEIFT, P. (1999a). Integrable systems and combinatorial theory. Notices Amer. Math. Soc. 47 631-640.
11. DEIFT, P. (1999b). Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Amer. Math. Soc., Providence, RI.
12. DUNSTER, T. M. (1989). Uniform asymptotic expansions for Whittaker's confluent hypergeometric functions. SIAM J. Math. Anal. 20 744-760.
13. DYSON, F. J. (1970). Correlations between eigenvalues of a random matrix. Comm. Math. Phys. 19 235-250.
14. EATON, M. L. (1983). Multivariate Statistics: A Vector Space Approach. Wiley, New York.
15. EDELMAN, A. (1988). Eigenvalues and condition numbers of random matrices. SIAM J. Matrix Anal. Appl. 9 543-560.
16. EDELMAN, A. (1991). The distribution and moments of the smallest eigenvalue of a random matrix of Wishart type. Linear Algebra Appl. 159 55-80.
17. ERDÉLYI, A. (1960). Asymptotic forms for Laguerre polynomials. J. Indian Math. Soc. 24 235-250.
18. FORRESTER, P. J. (1993). The spectrum edge of random matrix ensembles. Nuclear Phys. B 402 709-728.
19. FORRESTER, P. J. (2000). Painlevé transcendent evaluation of the scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles. Technical report. www.lanl.gov arXiv:nlin.SI/0005064.
20. GEMAN, S. (1980). A limit theorem for the norm of random matrices. Ann. Probab. 8 252-261.
21. GOHBERG, I. C. and KREIN, M. G. (1969). Introduction to the Theory of Linear Non-selfadjoint Operators. Amer. Math. Soc., Providence, RI.
22. HASTINGS, S. P. and MCLEOD, J. B. (1980). A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Rational Mech. Anal. 73 31-51.
23. HORN, R. A. and JOHNSON, C. R. (1985). Matrix Analysis. Cambridge Univ. Press.
24. JAMES, A. T. (1964). Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35 475-501.
25. JOHANSSON, K. (1998). On fluctations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 151-204.
26. JOHANSSON, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437-476.
27. MARDIA, K. V., KENT, J. T. and BIBBY, J. M. (1979). Multivariate Analysis. Academic Press, New York.
28. MARČENKO, V. A. and PASTUR, L. A. (1967). Distributions of eigenvalues of some sets of random matrices. Math. USSR-Sb. 1 507-536.
29. MEHTA, M. L. (1991). Random Matrices, 2nd ed. Academic Press, New York.
30. MUIRHEAD, R. J. (1974). Powers of the largest latent root test of Σ = I. Comm. Statist. 3 513-524.
31. MUIRHEAD, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
32. OLVER, F. W. J. (1974). Asymptotics and Special Functions. Academic Press, New York.
33. PREISENDORFER, R. W. (1988). Principal Component Analysis in Meteorology and Oceanogaphy. North-Holland, Amsterdam.
34. RIESZ, F. and SZ.-NAGY, B. (1955). Functional Analysis. Ungar, New York.
35. SOSHNIKOV, A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 697-733.
36. SOSHNIKOV, A. (2001). A note on universality of the distribution of the largest eigenvalues in certain classes of sample covariance matrices, Technical report, www.lanl.gov arXiv:math:PR/0104113.
37. STEIN, C. (1969). Multivariate analysis I. Technical report, Dept. Statistics Stanford Univ., pages 79-81. (Notes prepared by M. L. Eaton in 1966.)
38. SZEGÖ, G. (1967). Orthogonal Polynomials, 3rd ed. Amer. Math. Soc. Providence.
39. TEMME, N. M. (1990). Asymptotic estimates for Laguerre polynomials. J. Appl. Math. Phys. (ZAMP) 41 114-126.
40. TRACY, C. A. and WIDOM, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151-174.
41. TRACY, C. A. and WIDOM, H. (1996). On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 727-754.
42. TRACY, C. A. and WIDOM, H. (1998). Correlation functions, cluster functions, and spacing distributions for random matrices. J. Statis. Phys. 92 809-835.
43. TRACY, C. A. and WIDOM, H. (1999). Airy kernel and Painlevé II. Technical report, www.lanl.gov solv-int/9901004. To appear in CRM Proceedings and Lecture Notes: "Isomonodromic Deformations and Applications in Physics," J. Harnad, ed.
44. TRACY, C. A. and WIDOM, H. (2000). The distribution of the largest eigenvalue in the Gaussian ensembles. In Calogero-Moser-Sutherland Models (J. van Diejen and L. Vinet, eds.) 461-472. Springer, New York.
45. WACHTER, K. W. (1976). Probability plotting points for principal components. In Ninth Interface Symposium Computer Science and Statistics (D. Hoaglin and R. Welsch, eds.) 299-308. Prindle, Weber and Schmidt, Boston.
46. WIDOM, H. (1999). On the relation between orthogonal, symplectic and unitary ensembles. J. Statist. Phys. 94 347-363.
47. WIGNER, E. P. (1955). Characteristic vectors of bordered matrices of infinite dimensions. Ann. Math. 62 548-564.
48. WIGNER, E. P. (1958). On the distribution of the roots of certain symmetric matrices. Ann. Math. 67 325-328.
49. WILKS, S. S. (1962). Mathematical Statistics. Wiley, New York.