Open Access
2024 Projection inference for high-dimensional covariance matrices with structured shrinkage targets
Fabian Mies, Ansgar Steland
Electron. J. Statist. 18(1): 1643-1676 (2024). DOI: 10.1214/24-EJS2225


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 parameter. 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.

Funding Statement

The authors acknowledge support from Deutsche Forschungsgemeinschaft (DFG, grant STE 1034/11-2).


Download Citation

Fabian Mies. Ansgar Steland. "Projection inference for high-dimensional covariance matrices with structured shrinkage targets." Electron. J. Statist. 18 (1) 1643 - 1676, 2024.


Received: 1 October 2022; Published: 2024
First available in Project Euclid: 2 April 2024

Digital Object Identifier: 10.1214/24-EJS2225

Primary: 60F15 , 62G05
Secondary: 62M10

Keywords: bilinear form , Change-point , Gaussian approximation , non-stationary time series , shrinkage

Vol.18 • No. 1 • 2024
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