Open Access
April 2001 On the distribution of the largest eigenvalue in principal components analysis
Iain M. Johnstone
Ann. Statist. 29(2): 295-327 (April 2001). DOI: 10.1214/aos/1009210544


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 p­variate Wishart distribution on n degrees of freedom with identity covariance.

Consider the limit of large p and n with $n/p = \gamma \ge 1$. When centered by $\mu_p = (\sqrt{n-1} + \sqrt{p})^2$ and scaled by $\sigma_p = (\sqrt{n-1} + \sqrt{p})(1/\sqrt{n-1} + 1/\sqrt{p}^{1/3}$, the distribution of x(1) approaches the Tracey-Widom law of order 1, which is defined in terms of the Painlevé II differential equation and can be numerically evaluated and tabulated in 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.


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Iain M. Johnstone. "On the distribution of the largest eigenvalue in principal components analysis." Ann. Statist. 29 (2) 295 - 327, April 2001.


Published: April 2001
First available in Project Euclid: 24 December 2001

zbMATH: 1016.62078
MathSciNet: MR1863961
Digital Object Identifier: 10.1214/aos/1009210544

Primary: 62F20 , 62H25
Secondary: 33C45 , 60H25

Keywords: empirical orthogonal functions , Fredholm determinant , Karhunen–Loève transform , Laguerre ensemble , Laguerre polynomial , Largest eigenvalue , largest singular value , Liouville–Green method , Painlevé equation , Plancherel–Rotach asymptotics , Random matrix theory , Tracy–Widom distribution , Wishart distribution

Rights: Copyright © 2001 Institute of Mathematical Statistics

Vol.29 • No. 2 • April 2001
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