RMT4DS (version 0.0.1)

CovEst: Estimation of the Spectrum of Population Covariance Matrix

Description

Estimation of the eigenvalues of population covariance matrix given samples.

Usage

MPEst(X, n=nrow(X), k=1, num=NULL, penalty=FALSE, n_spike=0)
MomentEst(X, n=nrow(X), k=1, n_spike=0)

Value

MPEst and MomentEst give estimation of the spectrum of population covariance matrix and corresponding spectral density.

Arguments

X

n by p data matrix.

n

sample size.

k

repeated times in estimation. If k>1, estimation will be the average.

num

numbers of mass points chosen in estimation.

penalty

whether to implement L-1 penalty in inverting Marchenko-Pastur law

n_spike

number of spikes in population spectral.

Author

Xiucai Ding, Yichen Hu

Details

Given \(E(X)=0\) and \(Cov(X)=\Sigma\) with \(\Sigma\) unknown and fourth moment of X exists, we want to estimate spectrum of \(\Sigma\) from sample covariance matrix \(X'X/n\).

MPEst estimates spectrum by inverting Marchenko-Pastur law while MomentEst estimates spectrum by estimating the moment of population spectral density.

Those two functions give estimates of the eigenvalues by d and estimates of spectral density by xs and cdf.

References

[1] El Karoui, N. (2008). Spectrum estimation for large dimensional covariance matrices using random matrix theory. The Annals of Statistics, 36(6), 2757-2790.

[2] Kong, W., & Valiant, G. (2017). Spectrum estimation from samples. The Annals of Statistics, 45(5), 2218-2247.

Examples

require(MASS)
n = 500
p = 250
X = mvrnorm(n, rep(0,p), diag(c(rep(2,p/2),rep(1,p/2))))
MPEst(X, n)$d
MomentEst(X, n)$d