# Linear shrinkage estimator of the covariance matrix if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="BGP2014",package="HDShOP",cached_env=.Rdpack.currefs) The optimal linear shrinkage estimator of the covariance matrix that minimizes the Frobenius norm: [REMOVE_ME]$$ \hat{\Sigma}_{OLSE} = \hat{\alpha} S + \hat{\beta} \Sigma_0, [REMOVE_ME_2]$$ where $\hat{\alpha}$ and $\hat{\beta}$ are optimal shrinkage intensities given in Eq. (4.3) and (4.4) of if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="BGP2014;textual",package="HDShOP",cached_env=.Rdpack.currefs) . $S$ 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, $\frac{1}{p} I$. ```r CovShrinkBGP14(n, TM, SCM) ``` ## Arguments - `n`: sample size. - `TM`: the target matrix for the shrinkage estimator. - `SCM`: sample covariance matrix. ## Returns a list containing an object of class matrix (S) and the estimated shrinkage intensities $\hat{\alpha}$ and $\hat{\beta}$. ## 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 if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="BGP2014;textual",package="HDShOP",cached_env=.Rdpack.currefs) . $S$ 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, $\frac{1}{p} I$. ## Examples ```r # 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] ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs)

CovShrinkBGP14 function