## 1 Introduction  
Beyond a linear shrinkage estimation of a sample covariance matrix with another well-structured matrix, [PERSON] and [PERSON] [1, 2] proposed a nonlinear shrinkage method by taking into account eigenvalues of the population that are estimated from the eigenvalue distribution of a sample dataset. The original idea is from a random matrix theory which effectively deals with noise in large-dimensional covariance matrices; however, it requires complex mathematics and is difficult to understand for practitioners. Another simpler nonlinear method splits the sample dataset and directly uses eigenvalues on the validation subset as the population eigenvalues [3]. Although this method is less accurate, it is simpler and easier to understand than the previous nonlinear approach. Another extension of the linear shrinkage is using multiple target matrices. However, performances of the original linear shrinkage estimators depend on a chosen target matrix. Therefore, multiple different targets could avoid this dependency and make the estimator more flexible [4]. Instead of being more complex, the benefits of the shrinkage estimators are questioned by a simple approach of a portfolio of covariance estimators [5]. The portfolio of covariance estimators contains the sample covariance matrix and estimators with different assumption and direction error. Then, it simply takes an average of them to neutralize the estimation error of the sample covariance matrix by specification errors of the others. A general comparison of [PERSON] and [PERSON] [6] compares those simple portfolios with the linear shrinkage portfolios, combinations in this study could be equally weighted or even random weighted. They show that out-of-sample performances of the simple portfolios and the complex shrinkage portfolios perform within the same range, at least on a standard deviation of the portfolio returns. Therefore, they conclude that there is no statistical improvement and also no benefit of using those complicated shrinkage estimators. In practice, fund managers are not only looking for the best covariance estimator but also seeking alternatives in order to expand their fund's capacity. Those alternatives are different to the best one but have comparable portfolio performances. Therefore, the portfolio of estimators is a simple method but important for investment management and they also rise a question about the effectiveness of the shrinkage estimation. However, there is a lack of investigation on sustainability of those covariance matrix estimations. An advanced estimation which has highest performances but unsustainable outcomes is not preferable. For example, a portfolio changes its positions unreasonably just because of some outliers in the data, then it leads to other problems for the fund managers such as transaction costs or liquidity risk. Under the risk perspective, that is not a best portfolio and those alternatives are more attract to fund managers if they are more sustainable. In this study, we question the stability of multiple covariance matrices to understand how they work in the portfolio optimization under the data perspective and it helps fund managers to develop their sustainable investments. Firstly, does the strength of the shrinkage estimation come from the combination with other matrices or from an optimal shrinkage intensity calculation? Secondly, could we combine with other sample covariance matrices instead of using different matrices? The first question is important in theoretical mathematics because understanding characteristics of not only the sample covariance matrix but also the shrinkage estimation is a principal research topic in various domains. The second question is important in the practical finance because they prefer a simple, effective, understandable and explainable method to a new, complicated and unproven method. One problem with an estimated covariance matrix is highly fluctuating and sensitive with outliers [7; 8; 9; 10].