The global minimum-variance (GMV) portfolio derived from the sample covariance matrix often performs poorly due to large estimation errors. Linear shrinkage covariance estimators have been extensively studied to address this issue. This study proposes an optimal shrinkage intensity selection for the linear shrinkage estimator family using cross-validated negative log-likelihood function minimization. Moreover, we provide theoretical insights into the selection process. Empirical studies have shown that the proposed approach produces more stable covariance matrix estimators than the Frobenius loss minimization method, resulting in improved GMV portfolios. Furthermore, linear shrinkage estimators that use a diagonal matrix or a matrix based on a one-factor model as the target matrix generally achieve the best performance. They also outperform nonlinear shrinkage covariance estimators, especially with a large number of assets. This superiority is evident in terms of out-of-sample variance, turnover, and the Sharpe ratio.