Large covariance estimation by thresholding principal orthogonal complements

• 成果类型: Article
• 署名作者: Fan, Jianqing; Liao, Yuan; Mincheva, Martina
• 署名单位: Princeton University; University System of Maryland; University of Maryland College Park; Princeton University
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1111/rssb.12016
• 发表日期: 2013
• 页码: 603-680
• 关键词: dynamic-factor model; High-dimension; Matrix decomposition; PORTFOLIO SELECTION; components-analysis; LARGEST EIGENVALUE; false discovery; Optimal Rates; large number; Consistency

摘要

The paper deals with the estimation of a high dimensional covariance with a conditional sparsity structure and fast diverging eigenvalues. By assuming a sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors. We introduce the principal orthogonal complement thresholding method POET' to explore such an approximate factor structure with sparsity. The POET-estimator includes the sample covariance matrix, the factor-based covariance matrix, the thresholding estimator and the adaptive thresholding estimator as specific examples. We provide mathematical insights when the factor analysis is approximately the same as the principal component analysis for high dimensional data. The rates of convergence of the sparse residual covariance matrix and the conditional sparse covariance matrix are studied under various norms. It is shown that the effect of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the unobserved factors and their factor loadings are derived. The asymptotic results are also verified by extensive simulation studies. Finally, a real data application on portfolio allocation is presented.