Large Covariance Estimation by Thresholding Principal Orthogonal Complements

Jianqing Fan Princeton University Yuan Liao Princeton University Martina Mincheva University of Maryland

Statistics Theory and Methods mathscidoc:1806.33003

Journal of the Royal Statistical Society, 2013
This paper deals with the estimation of a high-dimensional covariance with a con- ditional sparsity structure and fast-diverging eigenvalues. By assuming 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 (POET) method to explore such an approximate factor structure with sparsity. The POET estimator includes the sample covariance matrix, the factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholding estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator (Cai and Liu, 2011) as speci c examples. We provide mathe- matical 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 impact of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the un- observed factors and their factor loadings are derived. The asymptotic results are also veri ed by extensive simulation studies. Finally, a real data application on portfolio allocation is presented.
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  title={Large Covariance Estimation by Thresholding Principal Orthogonal Complements},
  author={Jianqing Fan, Yuan Liao, and Martina Mincheva},
  booktitle={Journal of the Royal Statistical Society},
Jianqing Fan, Yuan Liao, and Martina Mincheva. Large Covariance Estimation by Thresholding Principal Orthogonal Complements. 2013. In Journal of the Royal Statistical Society.
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