WishartMax {RMTstat} | R Documentation |
Density, distribution function, quantile function, and random
generation for the maximum eigenvalue from a white Wishart matrix
(sample covariance matrix) with n.df
degrees of freedom,
p.dim
dimensions, population variance var
, and order
parameter beta
.
dwishart.max(x, n.df, p.dim, var=1, beta=1, log = FALSE) pwishart.max(q, n.df, p.dim, var=1, beta=1, lower.tail = TRUE, log.p = FALSE) qwishart.max(p, n.df, p.dim, var=1, beta=1, lower.tail = TRUE, log.p = FALSE) rwishart.max(n, n.df, p.dim, var=1, beta=1)
x,q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If length(n) > 1 , the length
is taken to be the number required. |
n.df |
the number of degrees of freedom for the Wishart matrix |
p.dim |
the number of dimensions (variables) for the Wishart matrix |
var |
the population variance. |
beta |
the order parameter (1 or 2) |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]. |
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 ntimes 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
n.df
and p.dim
both tend to infinity.
Supported values for beta
are 1
for real data and
and 2
for complex data.
dwishart.max
gives the density,
pwishart.max
gives the distribution function,
qwishart.max
gives the quantile function, and
rwishart.max
generates random deviates.
Patrick O. Perry
The functions are calculated by applying the appropriate centering and scaling (determined by wishart.max.par), and then calling the corresponding functions for the TracyWidom distribution.
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.
wishart.max.par, WishartSpike, TracyWidom