The determinant of the covariance matrix for high-dimensional data plays an important role in statistical inference and decision. It has many real applications including statistical tests and information theory. Due to the statistical and computational challenges with high dimensionality, little work has been proposed in the literature for estimating the determinant of high-dimensional covariance matrix. In this paper, we estimate the determinant of the covariance matrix using some recent proposals for estimating high-dimensional covariance matrix. Specifically, we consider a total of eight covariance matrix estimation methods for comparison. Through extensive simulation studies, we explore and summarize some interesting comparison results among all compared methods. We also provide practical guidelines based on the sample size, the dimension, and the correlation of the data set for estimating the determinant of high-dimensional covariance matrix. Finally, from a perspective of the loss function, the comparison study in this paper may also serve as a proxy to assess the performance of the covariance matrix estimation.

中文翻译：

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高维协方差矩阵行列式估计方法的比较。

高维数据的协方差矩阵的行列式在统计推断和决策中起着重要作用。它具有许多实际应用，包括统计测试和信息论。由于高维的统计和计算挑战，文献中很少有工作来估计高维协方差矩阵的行列式。在本文中，我们使用一些估计高维协方差矩阵的最新建议来估计协方差矩阵的行列式。具体而言，我们考虑总共使用八种协方差矩阵估计方法进行比较。通过广泛的仿真研究，我们探索并总结了所有比较方法中的一些有趣的比较结果。我们还会根据样本量，尺寸，以及用于估计高维协方差矩阵行列式的数据集的相关性。最后，从损失函数的角度来看，本文的比较研究还可以作为评估协方差矩阵估计性能的代理。