The estimation of structured covariance matrix arises in many fields. An appropriate covariance structure not only improves the accuracy of covariance estimation but also increases the efficiency of mean parameter estimators in statistical models. In this paper, a novel statistical method is proposed, which selects the optimal Toeplitz covariance structure and estimates the covariance matrix, simultaneously. An entropy loss function with nonconvex penalty is employed as a matrix-discrepancy measure, under which the optimal selection of sparse or nearly sparse Toeplitz structure and the parameter estimators of covariance matrix are made, simultaneously, through its minimization. The cases of both low-dimensional ($p\le n$) and high-dimensional ($p>n$) covariance matrix estimation are considered. The resulting Toeplitz structured covariance estimators are guaranteed to be positive definite and consistent. Asymptotic properties are investigated and simulation studies are conducted, showing that very high accurate Toeplitz covariance structure estimation is made. The proposed method is then applied to practical data analysis, which demonstrates its good performance in covariance estimation in practice.

结构协方差矩阵的估计出现在许多领域。适当的协方差结构不仅可以提高协方差估计的准确性，而且可以提高统计模型中平均参数估计量的效率。本文提出了一种新的统计方法，该方法选择最佳的Toeplitz协方差结构并同时估计协方差矩阵。将具有非凸罚分的熵损失函数用作矩阵离散度量，在此同时，通过使其最小化来同时选择稀疏或接近稀疏的Toeplitz结构的最佳选择和协方差矩阵的参数估计量。低维的情况（$p\le ñ$）和高维度（$p>ñ$）考虑了协方差矩阵估计。保证得到的Toeplitz结构化协方差估计量是正定的和一致的。对渐近性质进行了研究并进行了仿真研究，结果表明可以进行非常精确的Toeplitz协方差结构估计。将该方法应用于实际数据分析，证明了其在协方差估计中的良好性能。