R: The White Wishart Maximum Eigenvalue Distributions

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 `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.

### See Also

WishartMaxPar, WishartSpike, TracyWidom

[Package RMTstat version 0.3 Index]