Difference-Based High-Dimensional Long-Run Covariance Matrix Estimation for Mean-shift Time Series

2026-03-17

AI Summary

This study addresses the severe bias in conventional long-run covariance estimation for high-dimensional time series when the mean is non-stationary. To tackle arbitrary forms of mean variation, the authors introduce, for the first time, a differencing-based approach to construct an initial estimator, which is then combined with hard thresholding, soft thresholding, and tapering techniques to achieve sparse covariance estimation. The theoretical analysis explicitly characterizes how the estimation error depends on the sparsity of the covariance matrix, the presence and location of mean change points, as well as the interplay between dimensionality and sample size. Under general weak dependence conditions, the method’s convergence rate is rigorously established. Numerical experiments demonstrate the superior performance of the proposed estimator in high-dimensional settings with time-varying means.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

We consider estimation of high-dimensional long-run covariance matrices for time series with nonconstant means, a setting in which conventional estimators can be severely biased. To address this difficulty, we propose a difference-based initial estimator that is robust to a broad class of mean variations, and combine it with hard thresholding, soft thresholding, and tapering to obtain sparse long-run covariance estimators for high-dimensional data. We derive convergence rates for the resulting estimators under general temporal dependence and time-varying mean structures, showing explicitly how the rates depend on covariance sparsity, mean variation, dimension, and sample size. Numerical experiments show that the proposed methods perform favorably in high dimensions, especially when the mean evolves over time.

Problem

• high-dimensional
• long-run covariance
• mean-shift
• time series
• nonconstant means

Innovation

• difference-based estimation
• long-run covariance
• high-dimensional time series
• mean-shift robustness
• thresholding