High dimensional variable selection and covariance matrix estimation

This thesis presents new results on two important statistical problems: high dimensional variable selection and high dimensional covariance matrix estimation. Challenges of these problems lie in the mathematical complexity and the fact that in many cases traditional ideas may no longer work well or even break down due to the high dimensionality. New procedures are proposed in the thesis for both problems and their theoretical properties are studied extensively using techniques like large deviations and results from random matrix theory. In particular, for the first problem, an innovative dimension reduction method Sure Independence Screening (SIS) has been proposed and it is shown to possess the sure screening property for even exponentially growing dimensionality. For the second problem, a factor model approach is taken to reduce dimensionality and to estimate the covariance matrix. Situations under which the factor approach increases performance substantially or marginally compared to the sample approach are identified by theoretical studies. The thesis contributes to the understanding of statistical challenges with high dimensionality.