GeneralMPLaw {RMT4DS}

General Marchenko-Pastur Distribution

Description

Density, distribution function, quantile function and random generation for the general Marchenko-Pastur distribution, the limiting distribution of empirical spectral measure for large Wishart matrices.

Usage

qgmp(p, ndf=NULL, pdim=NULL, svr=ndf/pdim, eigens=NULL, lower.tail=TRUE,
    log.p=FALSE, m=500)
rgmp(n, ndf=NULL, pdim=NULL, svr=ndf/pdim, eigens=NULL, m=500)
pgmp(q, ndf=NULL, pdim=NULL, svr=ndf/pdim, eigens=NULL, lower.tail=TRUE,
    log.p=FALSE, m=500)
dgmp(x, ndf=NULL, pdim=NULL, svr=ndf/pdim, eigens=NULL, log.p=FALSE, m=500)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observation.

m

number of points used in estimating density.

ndf

the number of degrees of freedom for the Wishart matrix.

pdim

the number of dimensions (variables) for the Wishart matrix.

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 \le x], otherwise, P[X > x].

eigens

input eigenvalues of population covariance matrix.

Details

Those functions work only for non-spiked part.

To achieve high accuracy of estimation, eigens should be large, like larger than 500.

In general Marchenko Pastur distributions, the support of density is the union of one or more intervals.

Value

dgmp gives the density,

pgmp gives the distribution function,

qgmp gives the quantile function,

rgmp generates random deviates,

Author(s)

Xiucai Ding, Yichen Hu

Source

If eigens is missing, functions from package RMTstat will be used to compute classical Marchenko-Pastur distribution.

References

[1] Knowles, A., & Yin, J. (2017). Anisotropic local laws for random matrices. Probability Theory and Related Fields, 169(1), 257-352.

[2] Bai, Z., & Yao, J. (2012). On sample eigenvalues in a generalized spiked population model. Journal of Multivariate Analysis, 106, 167-177.

[3] Ding, X. (2021). Spiked sample covariance matrices with possibly multiple bulk components. Random Matrices: Theory and Applications, 10(01), 2150014.

[4] Ding, X., & Trogdon, T. (2021). A Riemann–Hilbert approach to the perturbation theory for orthogonal polynomials: Applications to numerical linear algebra and random matrix theory. arXiv preprint arXiv:2112.12354.

Examples

N = 1000
M = 300
d = c(rep(3.8,M/3),rep(1.25,M/3),rep(0.25,M/3))
qgmp(0.5, ndf=N, pdim=M, eigens=d)
pgmp(3, ndf=N, pdim=M, eigens=d)
dgmp(2, ndf=N, pdim=M, eigens=d)
rgmp(2, ndf=N, pdim=M, eigens=d)