MarcenkoPastur: The Marcenko-Pastur Distribution

In RMTstat: Distributions, Statistics and Tests Derived from Random Matrix Theory

The Marcenko-Pastur Distribution

Description

Density, distribution function, quantile function and random generation for the Marčenko-Pastur distribution, the limiting distribution of the empirical spectral measure for a large white Wishart matrix.

Usage

dmp( x, ndf=NA, pdim=NA, var=1, svr=ndf/pdim, log = FALSE )
pmp( q, ndf=NA, pdim=NA, var=1, svr=ndf/pdim, lower.tail = TRUE, log.p = FALSE )
qmp( p, ndf=NA, pdim=NA, var=1, svr=ndf/pdim, lower.tail = TRUE, log.p = FALSE )
rmp( n, ndf=NA, pdim=NA, var=1, svr=ndf/pdim )

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.
svr	samples to variables ratio; the number of degrees of freedom per dimension.
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 concentration can either be given explicitly, or else computed from the given ndf and pdim. If var is not specified, it assumes the default of 1.

The Marčenko-Pastur law is the limit of the random probability measure which puts equal mass on all pdim eigenvalues of a normalized pdim-dimensional white Wishart matrix with ndf degrees of freedom and scale parameter diag(var, var, ..., var). It is assumed that ndf goes to infinity, and ndf/pdim goes to nonzero constant called the "samples-to-variables ratio" (svr).

Value

dmp gives the density, pmp gives the distribution function, qmp gives the quantile function, and rmp generates random deviates.

Author(s)

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

Source

Other than the density, these functions are relatively slow and imprecise.

The distribution function is computed with integrate. The quantiles are computed via bisection using uniroot. Random variates are generated using the inverse CDF.

References

Marčenko, V.A. and Pastur, L.A. (1967). Distribution of eigenvalues for some sets of random matrices. Sbornik: Mathematics 1, 457–483.

RMTstat documentation built on April 13, 2022, 1:07 a.m.