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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 \(n\times p\) 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 \(n/p=\gamma\geq 1\). When centered by \(\mu_p=(\sqrt{n-1}+\sqrt p)^2\) and scaled by \(\sigma_p=\break (\sqrt{n-1}+\sqrt p)(1/\sqrt{n-1}+1/\sqrt p)^{1/3}\), 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.

MSC:
62H25 Factor analysis and principal components; correspondence analysis
62H10 Multivariate distribution of statistics
15B52 Random matrices (algebraic aspects)
33E17 Painlevé-type functions
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
60F05 Central limit and other weak theorems
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