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.
翻译:分析依赖下的高度数据的大量样本是一个具有挑战性的统计问题,因为长时间序列可能有变化点,其中最重要的是平均值和边际共变差点,对此需要有效的测试。大型共变矩阵由于噪音累积而特别难以推断出大型共变矩阵,导致单数估计,相关测试的功率低。高维抽样共变异矩阵的独特性可以通过考虑与正常的、结构性更强的目标矩阵进行线性组合来克服。这种方法被称为缩水,目标矩阵通常为对等形式。在本文件中,我们考虑向结构化的、非参数性的可带或托普利茨类型非参数性估算器进行缩缩缩缩,分别是为了提高测试的准确性和统计能力,甚至是在非静止状态下进行。我们从高维度和托普利茨型测试中得出可行的双线性估算结果,这些近似使我们能够为高维度时间序列的边际差异结构结构的结构性断裂进行统计测试。我们考虑的是,我们考虑在不设对可谱或托普利茨型类型类型进行结构的结构性估算。我们从模拟的精确度测测算中可以显示其精确度的精确度测试。我们通过模拟测算的精确度评估的精确度,我们通过测算来显示其精确度评估的结果。我们如何显示其精确度。