  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.
