Estimating covariance structure in high dimensions

Many of scientific domains rely on extracting knowledge from high-dimensional data sets to provide insights into complex mechanisms underlying these data. Statistical modeling has become ubiquitous in the analysis of high dimensional data for exploring the large-scale gene regulatory networks in hope of developing better treatments for deadly diseases, in search of better understanding of cognitive systems, and in prediction of volatility in stock market in the hope of averting the potential risk. Statistical analysis in these high-dimensional data sets yields better results only if an estimation procedure exploits hidden structures underlying the data. This thesis develops flexible estimation procedures with provable theoretical guarantees for estimating the unknown covariance structures underlying data generating process. Of particular interest are procedures that can be used on high dimensional data sets where the number of samples n is much smaller than the ambient dimension p. Due to the importance of structure estimation, the methodology is developed for the estimation of both covariance and its inverse in parametric and as well in non-parametric framework.