WishartMax {RMTstat}R Documentation

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 n.df degrees of freedom, p.dim dimensions, population variance var, and order parameter beta.

Usage

dwishart.max(x, n.df, p.dim, var=1, beta=1, log = FALSE)
pwishart.max(q, n.df, p.dim, var=1, beta=1, lower.tail = TRUE, log.p = FALSE)
qwishart.max(p, n.df, p.dim, var=1, beta=1, lower.tail = TRUE, log.p = FALSE)
rwishart.max(n, n.df, p.dim, 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.
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 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].

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 ntimes 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 n.df and p.dim both tend to infinity.

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

Value

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

Author(s)

Patrick O. Perry

Source

The functions are calculated by applying the appropriate centering and scaling (determined by wishart.max.par), and then calling the corresponding functions for the TracyWidom distribution.

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.

See Also

wishart.max.par, WishartSpike, TracyWidom


[Package RMTstat version 0.1 Index]