## 2 Background  
### Linear Strinkage Estimation  
After a seminal theory of portfolio selection of [PERSON] [11], the computation of efficient portfolios is an area of interest in finance profession which aims to estimate optimal weights for assets. That uses the mean and the covariance of stock returns as input parameters; however, an estimation error of the expected returns are larger than the covariance estimation error [12]. Recent research studies suggest assuming equal expected returns for all stocks and using only the covariance matrix to estimate the portfolio offer the lowest risk: in other words, a global minimum variance portfolio (GMVP) [13]. The standard statistical estimation of the covariance matrix in the GMVP is the sample covariance estimation. While the mathematics of the GMVP and the sample covariance matrix is straightforward, several studies in the theory and industrial practice pointed out the disadvantages of the sample covariance matrix in portfolio trading. When a number of assets in our universe are greater than a number of historical returns, formally is when a number of dimensions are greater than a number of observations, the estimated sample covariance matrix from those data is not invertible and also ill-conditioned. Portfolio optimization in this context contains high estimation error and it turns out that the portfolio needs to adjust its positions more frequently which will increase the risk level of the portfolio and also decrease its profit. One approach to make the covariance matrix invertible in statistics is by imposing the matrix structure, such as a shrinkage technique that takes a weighted combination of two matrices, the sample covariance matrix and another invertible matrix [14]. The weight, also known as a shrinkage intensity, is optimized from a quadratic loss function on a given dataset without looking at the out-of-sample data. In this case, there is a unique \"optimal\" shrinkage intensity for one dataset. On the other hand, some recent studies [15; 16] split the dataset and look at validation subsets to evaluate the \"optimal\" shrinkage intensity by cross-validation method instead of optimization, it resulted in different \"optimal\" shrinkage intensities based on different pre-defined objectives. Given a dataset of \(N\) random variables and \(T\) observations representing \(T\) returns of \(N\) assets on a universe, the GMVP is formulated as follows:  
\[ \begin{split}\min_{\mathbf{w}}&\quad\mathbf{w}^{ \intercal}\mathbf{\Sigma}\mathbf{w}\\ \text{s.t.}&\quad\mathbf{w}^{\intercal}\mathbf{1}=1 \end{split}  \tag{1}\]  
where the \(\mathbf{\Sigma}^{N\times N}\) is the covariance matrix and the \(\mathbf{w}=\{w_{1},\cdots,w_{N}\}\) is a weight vector of \(N\) assets respectively. The well-known solution of Equation (1) is:  
\[\mathbf{w}=\frac{\mathbf{\Sigma}^{-1}\mathbf{1}}{\mathbf{1}^{\intercal} \mathbf{\Sigma}^{-1}\mathbf{1}}. \tag{2}\]  
The most common estimator, the sample covariance matrix estimator, not always be invertible and also ill-conditioned. Particularly in the financial industry, when the number of assets \(N\) is greater than the number of observations \(T\) (\(N>>T\)), the inverse of the sample covariance matrix amplifies estimation error. For the high dimensional covariance matrices, refs. [17; 18; 19] proposed to impose the structure of the covariance matrix by using the shrinkage technique. They combined linearly the sample covariance matrix with another well-conditioned structured matrix. Given a target matrix \(\mathbf{F}\), the linear shrinkage estimator is as follows:  
\[\widehat{\Sigma}=\delta\mathbf{F}+(1-\delta)\mathbf{S} \tag{3}\]  
in which, \(\mathbf{S}\) is the sample covariance matrix and \(\delta\in(0,1)\) is a shrinkage intensity. The optimal \(\delta\) is obtained by minimize the quadratic loss of \(|\mathbf{F}-\mathbf{S}|\).