WishartSpike: The Spiked Wishart Maximum Eigenvalue Distributions

Description

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).

Usage

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.

vector of probabilities.

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 \(P[X \le x]\), otherwise, \(P[X > x]\).

Value

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 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.