Density, distribution function, quantile function, and random
generation for the maximum eigenvalue from a spiked Wishart matrix
(sample covariance matrix) with ndf
degrees of freedom,
pdim
dimensions, and population covariance matrix
diag(spike+var,var,var,...,var)
.
dWishartSpike(x, spike, ndf=NA, pdim=NA, var=1, beta=1, log = FALSE)
pWishartSpike(q, spike, ndf=NA, pdim=NA, var=1, beta=1, lower.tail = TRUE, log.p = FALSE)
qWishartSpike(p, spike, ndf=NA, pdim=NA, var=1, beta=1, lower.tail = TRUE, log.p = FALSE)
rWishartSpike(n, spike, ndf=NA, pdim=NA, 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 value of the spike.
the number of degrees of freedom for the Wishart matrix.
the number of dimensions (variables) for the Wishart matrix.
the population (noise) 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, \(P[X > x]\).
dWishartSpike
gives the density,
pWishartSpike
gives the distribution function,
qWishartSpike
gives the quantile function, and
rWishartSpike
generates random deviates.
The spiked Wishart is a random sample covariance matrix from
multivariate normal data with ndf
observations in pdim
dimensions. The spiked Wishart has one population covariance eigenvalue
equal to spike+var
and the rest equal to var
. These
functions are related to the limiting distribution of the largest eigenvalue
from such a matrix when ndf
and pdim
both tending to
infinity, with their ratio tending to a nonzero constant.
For the spiked distribution to exist, spike
must be greater than
sqrt(pdim/ndf)*var
.
Supported values for beta
are 1
for real data and
and 2
for complex data.
Baik, J., Ben Arous, G., and P<U+00E9>ch<U+00E9>, S. (2005). Phase transition of the largest eigenvalue for non-null complex sample covariance matrices. Annals of Probability 33, 1643--1697.
Baik, J. and Silverstein, J. W. (2006). Eigenvalues of large sample covariance matrices of spiked population models. Journal of Multivariate Analysis 97, 1382-1408.
Paul, D. (2007). Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statistica Sinica. 17, 1617--1642.