Statistical Inference for High-Dimensional Covariance Structure
Hermann Otto Hirschfeld Lectures
Presented at Humboldt-Universität zu Berlin, October 2012
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dimension p can be much larger than the sample size n, classical methods and results based on fixed p and large n are no longer applicable.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 lectures, I will discuss some recent results on optimal and adaptive estimation of large covariance matrices under different settings. The results and technical analysis reveal in some cases new features that are quite different from the conventional nonparametric function estimation problems. Time permitting, I will also discuss some applications.
Papers:
- Cai, T. T. & Liu, W. (2011a). Adaptive thresholding for sparse covariance matrix estimation. J. American Statistical Association 106, 672-684.
- Cai, T. T. & Liu, W. (2011b). A direct estimation approach to sparse linear discriminant analysis. J. American Statistical Association 106, 1566-1577.
- Cai, T. T., Liu, W., & Zhou, H. (2016). Estimating sparse precision matrix: Optimal rates of convergence and adaptive estimation. The Annals of Statistics 44, 455-488.
- Cai, T. T. & Yuan, M. (2012). Adaptive covariance matrix estimation through block thresholding. The Annals of Statistics 40, 2014-2042.
- 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. (2012). Optimal rates of convergence for sparse covariance matrix estimation. The Annals of Statistics 40, 2389-2420.