Condition Number Regularized Covariance Estimation.
Condition Number Regularized Covariance Estimation.
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DOI：
10.1111/j.1467-9868.2012.01049.x
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发表时间：
2013-06-01
期刊：
影响因子：
--
通讯作者：
Rajaratnam B
中科院分区：
文献类型：
--
作者：
Won JH;Lim J;Kim SJ;Rajaratnam B
高维协方差矩阵的估计是一个困难的问题，有许多应用，是目前较大的统计社区的兴趣。在许多应用中，包括所谓的“大p小n”设置，协方差矩阵的估计不仅要求是可逆的，而且是良好的条件。虽然许多正则化方案试图做到这一点，但它们都没有直接解决病态问题。在本文中，我们提出了一个最大似然方法，直接目标是获得一个良好的条件估计。对协方差矩阵或其逆矩阵都没有稀疏性假设，从而使我们的程序更广泛地适用。我们证明了所提出的正则化方案是计算效率高，产生一种类型的Steinian收缩估计，并具有自然的贝叶斯解释。我们全面研究了正则化协方差估计器的理论特性，包括其正则化路径，并着手开发一种自适应确定所需正则化水平的方法。最后，我们证明了正则化估计的性能在决策理论的比较和金融投资组合优化设置。所提出的方法具有理想的性能，并可以作为一个有竞争力的程序，特别是当样本量很小，当一个良好的条件估计是必需的。
Estimation of high-dimensional covariance matrices is known to be a difficult problem, has many applications, and is of current interest to the larger statistics community. In many applications including so-called the “large p small n” setting, the estimate of the covariance matrix is required to be not only invertible, but also well-conditioned. Although many regularization schemes attempt to do this, none of them address the ill-conditioning problem directly. In this paper, we propose a maximum likelihood approach, with the direct goal of obtaining a well-conditioned estimator. No sparsity assumption on either the covariance matrix or its inverse are are imposed, thus making our procedure more widely applicable. We demonstrate that the proposed regularization scheme is computationally efficient, yields a type of Steinian shrinkage estimator, and has a natural Bayesian interpretation. We investigate the theoretical properties of the regularized covariance estimator comprehensively, including its regularization path, and proceed to develop an approach that adaptively determines the level of regularization that is required. Finally, we demonstrate the performance of the regularized estimator in decision-theoretic comparisons and in the financial portfolio optimization setting. The proposed approach has desirable properties, and can serve as a competitive procedure, especially when the sample size is small and when a well-conditioned estimator is required.
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