Description Usage Arguments Details Value Author(s) References See Also
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)
.
1 2 3 4  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)

x,q 
vector of quantiles. 
p 
vector of probabilities. 
n 
number of observations. If 
spike 
the value of the spike. 
ndf 
the number of degrees of freedom for the Wishart matrix. 
pdim 
the number of dimensions (variables) for the Wishart matrix. 
var 
the population (noise) 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]. 
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.
dWishartSpike
gives the density,
pWishartSpike
gives the distribution function,
qWishartSpike
gives the quantile function, and
rWishartSpike
generates random deviates.
Iain M. Johnstone, Zongming Ma, Patrick O. Perry and Morteza Shahram
Baik, J., Ben Arous, G., and Péché, S. (2005). Phase transition of the largest eigenvalue for nonnull 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, 13821408.
Paul, D. (2007). Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statistica Sinica. 17, 1617–1642.
WishartSpikePar, WishartMax
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