In this article, we propose a computationally efficient approach to estimate (large) p-dimensional covariance matrices of ordered (or longitudinal) data based on an independent sample of size n. To do this, we construct the estimator based on a k-band partial autocorrelation matrix with the number of bands chosen using an exact multiple hypothesis testing procedure. This approach is considerably faster than many existing methods and only requires inversion of (k + 1)-dimensional covariance matrices. The resulting estimator is positive definite as long as k < n (where p can be larger than n). We make connections between this approach and banding the Cholesky factor of the modified Cholesky decomposition of the inverse covariance matrix (Wu and Pourahmadi, 2003) and show that the maximum likelihood estimator of the k-band partial autocorrelation matrix is the same as the k-band inverse Cholesky factor. We evaluate our estimator via extensive simulations and illustrate the approach using high-dimensional sonar data.

中文翻译：

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有序数据的大协方差矩阵的计算高效分带以及与逆 Cholesky 因子分带的连接

在本文中，我们提出了一种计算效率高的方法，用于基于大小为 n 的独立样本来估计有序（或纵向）数据的（大）p 维协方差矩阵。为此，我们基于 k 波段偏自相关矩阵构建估计器，其中使用精确的多假设检验程序选择波段数。这种方法比许多现有方法快得多，并且只需要 (k + 1) 维协方差矩阵的求逆。只要 k < n（其中 p 可以大于 n），由此产生的估计量就是正定的。我们将这种方法与对逆协方差矩阵（Wu 和 Pourahmadi，2003) 并表明 k 波段偏自相关矩阵的最大似然估计量与 k 波段逆 Cholesky 因子相同。我们通过广泛的模拟评估我们的估计器，并使用高维声纳数据说明该方法。