Projection inference for high-dimensional covariance matrices with structured shrinkage targets
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结构化收缩目标下高维协方差矩阵的投影推断
- 研究问题
- 在高维、依赖和非平稳数据中，如何改善协方差估计及结构突变检验，并降低噪声累积和样本协方差奇异带来的困难？
- 方法与思路
- 作者将样本协方差向可带状化或 Toeplitz 型非参数目标收缩，推导双线性投影的高斯近似，并提出突变检验及数据驱动权重。
- 材料陈述的贡献
- 摘要称近似允许依赖与非平稳，并支持不限制维度的协方差突变检验；模拟报告收缩提高功效，特定协方差情形下估计表现更好。
- 阅读关注点
- 应核对依赖、非平稳与投影向量的条件，维度不受限的具体含义，以及权重选择、检验尺寸和功效比较所覆盖的模拟情形。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未获核验；摘要未列出近似成立条件，特定模拟优势不能推广至任意协方差结构。
原文摘要
Analyzing large samples of high-dimensional data under dependence is a challenging statistical problem as long time series may have change points, most importantly in the mean and the marginal covariances, for which one needs valid tests. Inference for large covariance matrices is especially difficult due to noise accumulation, resulting in singular estimates and poor power of related tests. The singularity of the sample covariance matrix in high dimensions can be overcome by considering a linear combination with a regular, more structured target matrix. This approach is known as shrinkage, and the target matrix is typically of diagonal form. In this paper, we consider covariance shrinkage towards structured nonparametric estimators of the bandable or Toeplitz type, respectively, aiming at improved estimation accuracy and statistical power of tests even under nonstationarity. We derive feasible Gaussian approximation results for bilinear projections of the shrinkage estimators which are valid under nonstationarity and dependence. These approximations especially enable us to formulate a statistical test for structural breaks in the marginal covariance structure of high-dimensional time series without restrictions on the dimension, and which is robust against nonstationarity of nuisance parameters. We show via simulations that shrinkage helps to increase the power of the proposed tests. Moreover, we suggest a data-driven choice of the shrinkage weights, and assess its performance by means of a Monte Carlo study. The results indicate that the proposed shrinkage estimator is superior for non-Toeplitz covariance structures close to fractional Gaussian noise.