# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On the distribution of the largest eigenvalue in principal components analysis. (English) Zbl 1016.62078

Summary: Let ${x}_{\left(1\right)}$ denote the square of the largest singular value of an $n×p$ matrix $X$, all of whose entries are independent standard Gaussian variates. Equivalently, ${x}_{\left(1\right)}$ is the largest principal component variance of the covariance matrix ${X}^{\text{'}}X$, 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 $n/p=\gamma \ge 1$. When centered by ${\mu }_{p}={\left(\sqrt{n-1}+\sqrt{p}\right)}^{2}$ and scaled by ${\sigma }_{p}=\left(\sqrt{n-1}+\sqrt{p}\right){\left(1/\sqrt{n-1}+1/\sqrt{p}\right)}^{1/3}$, the distribution of ${x}_{\left(1\right)}$ 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 distributions of statistics 15A52 Random matrices (MSC2000) 33E17 Painlevé-type functions 33C45 Orthogonal polynomials and functions of hypergeometric type 60F05 Central limit and other weak theorems