Variance adaptation and covariance regularization in sparse inference

Modern statistical research focuses on problems in high-dimensional data analysis. This thesis focuses on two interrelated topics: (1) signal detection in sparse normal mixtures; and (2) variable selection in sparse regression models. Both topics consider the sparse feature of high-dimensional data. Much of the statistical challenge for the aforementioned topics is caused by the large-scale covariance matrix. In order to incorporate the covariance information properly, covariance adaptation and regularization are studied in this thesis. For signal detection problem, the effect of signal variance on the detectable region is derived explicitly and a non-parametric method called Higher Criticism is studied and proved to be optimally adaptive to unknown signal variance. For variable selection in regression, several new variable selection methods are introduced to utilize the covariance information through covariance regularization. This thesis presents in detail the methodologies, theoretical analysis and some simulation examples. More simulations and real data analysis can be found in Jeng and Jin (2009), Jeng et al. (2008) and Daye and Jeng (2009)