标题： 正则化的大协方差矩阵估计 显示英文标题

标题： Regularized estimation of large covariance matrices

摘要： 本文研究了通过带状化或 tapered 方法估计来自 $n$ 个观测值的 $p$ 个变量的协方差矩阵，或者估计协方差矩阵逆的一个带状版本。 我们证明了只要 $(\log p)/n\to0$，这些估计在算子范数下是一致的，并且得到了明确的速率。 结果对于某些相当自然且条件良好的协方差矩阵族是一致的。 我们还引入了一个类似于高斯白噪声模型的类比，并表明如果总体协方差可以嵌入该模型并且条件良好，则带状近似能够一致地估计协方差矩阵的特征值及其对应的特征向量。 这些结果可以推广到带状化的平滑版本以及具有足够短尾部的非高斯分布。 提出了一种重抽样方法来在实际中选择带宽参数。 这种方法在模拟数据和真实数据上进行了数值验证。 显示英文摘要

摘要： This paper considers estimating a covariance matrix of $p$ variables from $n$ observations by either banding or tapering the sample covariance matrix, or estimating a banded version of the inverse of the covariance. We show that these estimates are consistent in the operator norm as long as $(\log p)/n\to0$, and obtain explicit rates. The results are uniform over some fairly natural well-conditioned families of covariance matrices. We also introduce an analogue of the Gaussian white noise model and show that if the population covariance is embeddable in that model and well-conditioned, then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix. The results can be extended to smooth versions of banding and to non-Gaussian distributions with sufficiently short tails. A resampling approach is proposed for choosing the banding parameter in practice. This approach is illustrated numerically on both simulated and real data.