### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 603-680 |

Number of pages | 78 |

Journal | Journal of the Royal Statistical Society. Series B: Statistical Methodology |

Volume | 75 |

Issue number | 4 |

DOIs | |

State | Published - Sep 2013 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Approximate factor model
- Cross-sectional correlation
- Diverging eigenvalues
- High dimensionality
- Low rank matrix
- Principal components
- Sparse matrix
- Thresholding
- Unknown factors

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## Cite this

*Journal of the Royal Statistical Society. Series B: Statistical Methodology*,

*75*(4), 603-680. https://doi.org/10.1111/rssb.12016