The Marcenko-Pastur Distribution
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.
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 )
x,q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
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]. |
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).
dmp gives the density,
pmp gives the distribution function,
qmp gives the quantile function, and
rmp generates random deviates.
Iain M. Johnstone, Zongming Ma, Patrick O. Perry and Morteza Shahram
Other than the density, these functions are relatively slow and imprecise.
Marčenko, V.A. and Pastur, L.A. (1967). Distribution of eigenvalues for some sets of random matrices. Sbornik: Mathematics 1, 457–483.
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