GeneralWM {RMT4DS} | R Documentation |
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)
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
|
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.
Value
dWishartMax
gives the density,
pWishartMax
gives the distribution function,
qWishartMax
gives the quantile function,
rWishartMax
generates random deviates.
Author(s)
Xiucai Ding, Yichen Hu
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)