Large Covariance Estimation by Thresholding Principal Orthogonal Complements

• Activity Number: 45
• Type: Contributed
• Date/Time: Sunday, August 4, 2013 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Martina Mincheva*+ and Jianqing Fan and Yuan Liao
• Companies: Princeton University and Princeton University and University of Maryland
• Keywords: High dimensionality ; approximate factor model ; sparse matrix ; principal components ; thresholding ; cross-sectional correlation

Abstract

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 the presence of the cross- sectional correlation even after taking out common but unobservable factors. We intro- duce 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 es- timator (Cai and Liu, 2011) as specic 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 impact of estimating the unknown factors vanishes as the dimen- sionality increases.