Estimation of a large sparse covariance matrix is of great importance for statistical analysis, especially in high-dimensional settings. The traditional approach such as the sample covariance matrix performs poorly due to the high dimensionality. The modified Cholesky decomposition (MCD) is a commonly used method for sparse covariance matrix estimation. However, the MCD method relies on the order of variables, which often is not available or cannot be pre-determined in practice. In this work, we solve this order issue by obtaining a set of covariance matrix estimates based on assuming different orders of variables used in the MCD. Then we consider an ensemble estimator as the “centre” of such a set of covariance matrix estimates with respect to the Frobenius norm. Our proposed method not only ensures that the estimator is positive definite, but also captures the underlying sparse structure of the covariance matrix. Under some regularity conditions, we establish both algorithmic and asymptotic convergence of the proposed method. Its merits are illustrated via simulation studies and a practical example using data from a prostate cancer study.

中文翻译：

---

基于 Cholesky 估计的稀疏协方差矩阵的变量排序

估计大型稀疏协方差矩阵对于统计分析非常重要，尤其是在高维设置中。样本协方差矩阵等传统方法由于维数高而表现不佳。修正的 Cholesky 分解 (MCD) 是一种常用的稀疏协方差矩阵估计方法。然而，MCD 方法依赖于变量的顺序，这在实践中通常不可用或无法预先确定。在这项工作中，我们通过假设 MCD 中使用的变量的不同顺序获得一组协方差矩阵估计来解决这个顺序问题。然后，我们将集合估计器视为此类相对于Frobenius范数的一组协方差矩阵估计的“中心”。我们提出的方法不仅确保估计量是正定的，但也捕获了协方差矩阵的底层稀疏结构。在一定的规律性条件下，我们建立了所提出方法的算法收敛性和渐近收敛性。它的优点通过模拟研究和一个使用前列腺癌研究数据的实际例子来说明。