CovEst {RMT4DS}

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)

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.

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.

Value

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

Author(s)

Xiucai Ding, Yichen Hu

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