v2-estimation for smooth eigenvectors of matrix-valued functions

• 成果类型: Article
• 署名作者: Motta, Giovanni; Wu, Wei Biao; Pourahmadi, Mohsen
• 署名单位: Texas A&M University System; Texas A&M University College Station; University of Chicago
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asad018
• 发表日期: 2023
• 页码: 10771098
• 关键词: sample autocovariance matrices; factor models; time-series; covariance; identification

摘要

Modern statistical methods for multivariate time series rely on the eigendecomposition of matrix-valued functions such as time-varying covariance and spectral density matrices. The curse of indeterminacy or misidentification of smooth eigenvector functions has not received much attention. We resolve this important problem and recover smooth trajectories by examining the distance between the eigenvectors of the same matrix-valued function evaluated at two consecutive points. We change the sign of the next eigenvector if its distance with the current one is larger than the square root of 2. In the case of distinct eigenvalues, this simple method delivers smooth eigenvectors. For coalescing eigenvalues, we match the corresponding eigenvectors and apply an additional signing around the coalescing points. We establish consistency and rates of convergence for the proposed smooth eigenvector estimators. Simulation results and applications to real data confirm that our approach is needed to obtain smooth eigenvectors.