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This paper deals with the estimation of a high-dimensional covariance with aconditional sparsity structure and fast-diverging eigenvalues. By assumingsparse error covariance matrix in an approximate factor model, we allow for thepresence of some cross-sectional correlation even after taking out common butunobservable factors. We introduce the Principal Orthogonal complEmentThresholding (POET) method to explore such an approximate factor structure withsparsity. The POET estimator includes the sample covariance matrix, thefactor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholdingestimator (Bickel and Levina, 2008) and the adaptive thresholding estimator(Cai and Liu, 2011) as specific examples. We provide mathematical insights whenthe factor analysis is approximately the same as the principal componentanalysis for high-dimensional data. The rates of convergence of the sparseresidual covariance matrix and the conditional sparse covariance matrix arestudied under various norms. It is shown that the impact of estimating theunknown factors vanishes as the dimensionality increases. The uniform rates ofconvergence 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.

Large Covariance Estimation by Thresholding Principal Orthogonal Complements