Density, distribution function, quantile function, and random
generation for the maximum eigenvalue from a white Wishart matrix
(sample covariance matrix) with ndf
degrees of freedom,
pdim
dimensions, population variance var
, and order
parameter beta
.
dWishartMax(x, ndf, pdim, var=1, beta=1, log = FALSE)
pWishartMax(q, ndf, pdim, var=1, beta=1, lower.tail = TRUE, log.p = FALSE)
qWishartMax(p, ndf, pdim, var=1, beta=1, lower.tail = TRUE, log.p = FALSE)
rWishartMax(n, ndf, pdim, var=1, beta=1)
vector of quantiles.
vector of probabilities.
number of observations. If length(n) > 1
, the length
is taken to be the number required.
the number of degrees of freedom for the Wishart matrix
the number of dimensions (variables) for the Wishart matrix
the population variance.
the order parameter (1 or 2).
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).
dWishartMax
gives the density,
pWishartMax
gives the distribution function,
qWishartMax
gives the quantile function, and
rWishartMax
generates random deviates.
If beta
is not specified, it assumes the default value of 1
.
Likewise, var
assumes a default of 1
.
A white Wishart matrix is equal in distribution to \( (1/n) X' X \),
where \(X\) is an \(n\times p\) matrix with elements i.i.d. Normal
with mean zero and variance var
. These functions give the limiting
distribution of the largest eigenvalue from the such a matrix when
ndf
and pdim
both tend to infinity.
Supported values for beta
are 1
for real data and
and 2
for complex data.
Johansson, K. (2000). Shape fluctuations and random matrices. Communications in Mathematical Physics. 209 437--476.
Johnstone, I.M. (2001). On the ditribution of the largest eigenvalue in principal component analysis. Annals of Statistics. 29 295--327.