RMT4DS (version 0.0.1)

GeneralWM: The Wishart Maximum Eigenvalue Distribution

Description

Density, distribution function, quantile function and random generation for the maximum eigenvalue from a general non-spiked Wishart matrix (sample covariance matrix) with ndf degrees of freedom, pdim dimensions, and order parameter beta.

Usage

dWishartMax(x, eigens, ndf, pdim, beta, log = FALSE)
pWishartMax(q, eigens, ndf, pdim, beta, lower.tail = TRUE, log.p = FALSE)
qWishartMax(p, eigens, ndf, pdim, beta, lower.tail = TRUE, log.p = FALSE)
rWishartMax(n, eigens, ndf, pdim, beta)

Value

dWishartMax gives the density,

pWishartMax gives the distribution function,

qWishartMax gives the quantile function,

rWishartMax generates random deviates.

Arguments

x,q

vector of quantiles.

p

vector of probabilities.

n

number of observations.

eigens

eigenvalues of population covariance matrix.

ndf

the number of degrees of freedom for the Wishart matrix

pdim

the number of dimensions (variables) for the Wishart matrix

beta

the order parameter. 1 for real Wishart and 2 for complex Wishart.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).

Author

Xiucai Ding, Yichen Hu

Details

A real Wishart matrix is equal in distribution to \(X^T X/n\), where \(X\) are \(n\times p\) real matrix with elements of mean zero and covariance matrix \(\Sigma\). A complex Wishart matrix is equal in distribution to \(X^* X/n\), where both real and imagety part of \(X\) are \(n\times p\) complex matrice with elements of mean zero and covariance matrix \(\Sigma/2\). eigens are eigenvalues of \(\Sigma\). These functions give the limiting distribution of the largest eigenvalue from the such a matrix when ndf and pdim both tend to infinity.

References

[1] El Karoui, N. (2007). Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices. The Annals of Probability, 35(2), 663-714.

[2] Lee, J. O., & Schnelli, K. (2016). Tracy–Widom distribution for the largest eigenvalue of real sample covariance matrices with general population. The Annals of Applied Probability, 26(6), 3786-3839.

Examples

n = 500
p = 100
eigens = c(rep(2,p/2), rep(1, p/2))
beta = 2
rWishartMax(5, eigens, n, p, beta=beta)
qWishartMax(0.5, eigens, n, p, beta=beta)
pWishartMax(3.5, eigens, n, p, beta=beta)
dWishartMax(3.5, eigens, n, p, beta=beta)