RMTstat: WishartMax – R documentation

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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.
`p`	vector of probabilities.
`n`	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 <= 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 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.

Value

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

Author(s)

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

Source

The functions are calculated by applying the appropriate centering and scaling (determined by WishartMaxPar), 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.