WishartSpike {RMTstat}R Documentation

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 n.df degrees of freedom, p.dim dimensions, and population covariance matrix diag(spike+var,var,var,...,var).

Usage

dwishart.spike(x, spike, n.df=NA, p.dim=NA, var=1, log = FALSE)
pwishart.spike(q, spike, n.df=NA, p.dim=NA, var=1, lower.tail = TRUE, log.p = FALSE)
qwishart.spike(p, spike, n.df=NA, p.dim=NA, var=1, lower.tail = TRUE, log.p = FALSE)
rwishart.spike(n, spike, n.df=NA, p.dim=NA, var=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.
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 (noise) variance.
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].

Details

The spiked Wishart is a random sample covariance matrix from multivariate normal data with n.df observations in p.dim 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 n.df and p.dim both tending to infinity, with their ratio tending to a nonzero constant.

For the spiked distribution to exist, spike must be greater than sqrt(p.dim/n.df)*var.

Value

dwishart.spike gives the density, pwishart.spike gives the distribution function, qwishart.spike gives the quantile function, and rwishart.spike generates random deviates.

Author(s)

Patrick O. Perry

References

Baik, J., Ben Arous, G., and Péché, 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.

See Also

wishart.spike.par, WishartMax


[Package RMTstat version 0.1 Index]