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dimension p can be much larger than the sample size n, classical methods and results based on fixed p and large n are no longer applicable.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.




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