InvCovShrinkBGP16 {HDShOP}

Linear shrinkage estimator of the inverse covariance matrix (Bodnar et al. 2016)

Description

The optimal linear shrinkage estimator of the inverse covariance (precision) matrix that minimizes the Frobenius norm is given by:

\hat{\Pi}_{OLSE} = \hat{\alpha} \hat{\Pi} + \hat{\beta} \Pi_0,

where \hat{\alpha} and \hat{\beta} are optimal shrinkage intensities given in Eq. (4.4) and (4.5) of Bodnar et al. (2016). \hat{\Pi} is the inverse of the sample covariance matrix (iSCM) and \Pi_0 is a positive definite symmetric matrix used as the target matrix (TM), for example, I.

Usage

InvCovShrinkBGP16(n, p, TM, iSCM)

Arguments

n

the number of observations

p

the number of variables (rows of the covariance matrix)

TM

the target matrix for the shrinkage estimator

iSCM

the inverse of the sample covariance matrix

Value

a list containing an object of class matrix (S) and the estimated shrinkage intensities \hat{\alpha} and \hat{\beta}.

References

Bodnar T, Gupta AK, Parolya N (2016). “Direct shrinkage estimation of large dimensional precision matrix.” Journal of Multivariate Analysis, 146, 223–236.

Examples

# Parameter setting
n <- 3e2
c <- 0.7
p <- c*n
mu <- rep(0, p)
Sigma <- RandCovMtrx(p=p)

# Generating observations
X <- t(MASS::mvrnorm(n=n, mu=mu, Sigma=Sigma))

# Estimation
TM <- matrix(0, nrow=p, ncol=p)
diag(TM) <- 1
iSCM <- solve(Sigma_sample_estimator(X))
Sigma_shr <- InvCovShrinkBGP16(n=n, p=p, TM=TM, iSCM=iSCM)
Sigma_shr$S[1:6, 1:6]