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On the distribution of the largest eigenvalue in principal components analysis. (English) Zbl 1016.62078

Summary: Let x (1) denote the square of the largest singular value of an n×p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component variance of the covariance matrix X ' X, or the largest eigenvalue of a p-variate Wishart distribution with n degrees of freedom and identity covariance. Consider the limit of large p and n with n/p=γ1. When centered by μ p =(n-1+p) 2 and scaled by σ p =(n-1+p)(1/n-1+1/p) 1/3 , the distribution of x (1) approaches the Tracy-Widom law [C.A. Tracy and H. Widom, J. Stat. Phys. 92, No. 5-6, 809-835 (1998; Zbl 0942.60099)] of order 1, which is defined in terms of a Painlevé II differential equation and can be numerically evaluated and tabulated by software.

Simulations show the approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts.

62H25Factor analysis and principal components; correspondence analysis
62H10Multivariate distributions of statistics
15A52Random matrices (MSC2000)
33E17Painlevé-type functions
33C45Orthogonal polynomials and functions of hypergeometric type
60F05Central limit and other weak theorems