WishartMax: The White Wishart Maximum Eigenvalue Distributions

Description

Density, distribution function, quantile function, and random generation for the maximum eigenvalue from a white Wishart matrix (sample covariance matrix) with ndf degrees of freedom, pdim dimensions, population variance var, and order parameter beta.

Usage

dWishartMax(x, ndf, pdim, var=1, beta=1, log = FALSE)
pWishartMax(q, ndf, pdim, var=1, beta=1, lower.tail = TRUE, log.p = FALSE)
qWishartMax(p, ndf, pdim, var=1, beta=1, lower.tail = TRUE, log.p = FALSE)
rWishartMax(n, ndf, pdim, 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.

ndf

the number of degrees of freedom for the Wishart matrix

pdim

the number of dimensions (variables) for the Wishart matrix

var

the population 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

dWishartMax gives the density, pWishartMax gives the distribution function, qWishartMax gives the quantile function, and rWishartMax generates random deviates.

Details

If beta is not specified, it assumes the default value of 1. Likewise, var assumes a default of 1.

A white Wishart matrix is equal in distribution to \( (1/n) X' X \), where \(X\) is an \(n\times p\) matrix with elements i.i.d. Normal with mean zero and variance var. These functions give the limiting distribution of the largest eigenvalue from the such a matrix when ndf and pdim both tend to infinity.

Supported values for beta are 1 for real data and and 2 for complex data.

References

Johansson, K. (2000). Shape fluctuations and random matrices. Communications in Mathematical Physics. 209 437--476.

Johnstone, I.M. (2001). On the ditribution of the largest eigenvalue in principal component analysis. Annals of Statistics. 29 295--327.