The Annals of Statistics

On the distribution of the largest eigenvalue in principal components analysis

Iain M. Johnstone

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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.

Article information

Ann. Statist., Volume 29, Number 2 (2001), 295-327.

First available in Project Euclid: 24 December 2001

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H25: Factor analysis and principal components; correspondence analysis 62F20
Secondary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 60H25: Random operators and equations [See also 47B80]

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


Johnstone, Iain M. On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 (2001), no. 2, 295--327. doi:10.1214/aos/1009210544.

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