We study high‐dimensional covariance/precision matrix estimation under the assumption that the covariance/precision matrix can be decomposed into a low‐rank component $L$ and a diagonal component $D$. The rank of $L$ can either be chosen to be small or controlled by a penalty function. Under moderate conditions on the population covariance/precision matrix itself and on the penalty function, we prove some consistency results for our estimators. A block‐wise coordinate descent algorithm, which iteratively updates $L$ and $D$, is then proposed to obtain the estimator in practice. Finally, various numerical experiments are presented; using simulated data, we show that our estimator performs quite well in terms of the Kullback–Leibler loss; using stock return data, we show that our method can be applied to obtain enhanced solutions to the Markowitz portfolio selection problem. The Canadian Journal of Statistics 48: 308–337; 2020 © 2019 Statistical Society of Canada

中文翻译：

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使用低秩和对角分解的高维协方差矩阵估计

我们在假设协方差/精度矩阵可以分解为低阶分量的假设下研究高维协方差/精度矩阵估计 $\mathrm{大号}$ 和一个对角线分量 $d$。的等级$\mathrm{大号}$可以选择较小或由惩罚函数控制。在总体条件下，总体协方差/精度矩阵本身和惩罚函数，我们证明了估计量的一致性。逐块坐标下降算法，该算法迭代更新$\mathrm{大号}$ 和 $d$然后提出，以在实践中获得估计量。最后，给出了各种数值实验。使用模拟数据，我们证明我们的估计器在Kullback-Leibler损失方面表现良好；使用股票收益数据，我们证明了我们的方法可以用于获得Markowitz投资组合选择问题的增强解。加拿大统计杂志48：308-337；加拿大 2020©2019加拿大统计学会