Statistical Inference for High-Dimensional Covariance Structure
Forum Lecture
Presented at the European Meeting of Statisticians 2010 in Priraeus, Greece
Abstract: Covariance structure is of fundamental importance in many areas of statistical inference and a wide range of applications, including genomics, fMRI analysis, risk management, and web search problems. In the high dimensional setting where the dimension p can be much larger than the sample size n, classical methods and results based on fixed p and large n are no longer applicable. In these two talks, I will discuss some new results on optimal estimation of large covariance matrices under different settings. The results and technical analysis reveal new features that are quite different from the conventional nonparametric function estimation problems. I will also discuss optimal estimation of a sparse precision matrix which has close connections to graphical model selection. We will introduce a constrained l1 minimization method for sparse precision matrix estimation and discuss its optimality. In addition, I will also discuss testing of covariance structure in the high dimensional setting based on recent results from random matrix theory.
Papers:
- Cai, T. T. & Jiang, T. (2011). Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices. The Annals of Statistics 39, 1496-1525.
- Cai, T. T., Liu, W. & Luo, X. (2011). A constrained l1 minimization approach to sparse precision matrix estimation. J. American Statistical Association 106, 594-607.
- Cai, T. T. & Liu, W. (2011). Adaptive thresholding for sparse covariance matrix estimation. J. American Statistical Association 106, 672-684.
- Cai, T. T., Wang, L. & Xu, G. (2010). Stable recovery of sparse signals and an oracle inequality. IEEE Transactions on Information Theory 56, 3516-3522.
- Cai, T. T., Zhang, C.-H. & Zhou, H. (2010). Optimal rates of convergence for covariance matrix estimation. The Annals of Statistics 38, 2118-2144.
- Cai, T. T. & Zhou, H. (2010). Optimal rates of convergence for sparse covariance matrix estimation. Technical report.
- Cai, T. T. & Zhou, H. (2010). Minimax estimation of large covariance matrices under l1 norm. Technical Report.
- Cai, T. T., Ren, Z. and Zhou, H. (2010). Optimal estimation of large Toeplitz covariance matrices. Manuscript.