CovShrinkBGP14 {HDShOP}

Linear shrinkage estimator of the covariance matrix (Bodnar et al. 2014)

Description

The optimal linear shrinkage estimator of the covariance matrix that minimizes the Frobenius norm:

\hat{\Sigma}_{OLSE} = \hat{\alpha} S + \hat{\beta} \Sigma_0,

where \hat{\alpha} and \hat{\beta} are optimal shrinkage intensities given in Eq. (4.3) and (4.4) of Bodnar et al. (2014). S is the sample covariance matrix (SCM, see Sigma_sample_estimator) and \Sigma_0 is a positive definite symmetric matrix used as the target matrix (TM), for example, \frac{1}{p} I.

Usage

CovShrinkBGP14(n, TM, SCM)

Arguments

n

sample size.

TM

the target matrix for the shrinkage estimator.

SCM

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 (2014). “On the strong convergence of the optimal linear shrinkage estimator for large dimensional covariance matrix.” Journal of Multivariate Analysis, 132, 215–228.

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/p
SCM <- Sigma_sample_estimator(X)
Sigma_shr <- CovShrinkBGP14(n=n, TM=TM, SCM=SCM)
Sigma_shr$S[1:6, 1:6]