# Linear shrinkage estimator of the inverse covariance matrix if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="BGP2016",package="HDShOP",cached_env=.Rdpack.currefs) The optimal linear shrinkage estimator of the inverse covariance (precision) matrix that minimizes the Frobenius norm is given by: [REMOVE_ME]$$ \hat{\Pi}_{OLSE} = \hat{\alpha} \hat{\Pi} + \hat{\beta} \Pi_0, [REMOVE_ME_2]$$ where $\hat{\alpha}$ and $\hat{\beta}$ are optimal shrinkage intensities given in Eq. (4.4) and (4.5) of if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="BGP2016;textual",package="HDShOP",cached_env=.Rdpack.currefs) . $\hat{\Pi}$ 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. ```r 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 ## 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 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 if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="BGP2016;textual",package="HDShOP",cached_env=.Rdpack.currefs) . $\hat{\Pi}$ 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. ## 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 iSCM <- solve(Sigma_sample_estimator(X)) Sigma_shr <- InvCovShrinkBGP16(n=n, p=p, TM=TM, iSCM=iSCM) Sigma_shr$S[1:6, 1:6] ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs)

InvCovShrinkBGP16 function