WishartSpike() R function from [RMTstat]

The Spiked Wishart Maximum Eigenvalue Distributions

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)

Arguments

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.
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, $P[X > x]$ .

Returns

dWishartSpike gives the density, pWishartSpike gives the distribution function, qWishartSpike gives the quantile function, and rWishartSpike generates random deviates.

Details

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.

References

Baik, J., Ben Arous, G., and , 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.

Author(s)

Iain M. Johnstone, Zongming Ma, Patrick O. Perry and Morteza Shahram

WishartSpike function