Statistical Guarantees of Tuning-Free Methods for Gaussian Graphical Models

The majority of methods for sparse precision matrix estimation rely on computationally expensive procedures, such as cross-validation, to determine the proper level of regularization.Recently, a special case of precision matrix estimation based on a distributionally robust optimization (DRO) framework has been shown to be equivalent to the graphical lasso. From this formulation, a method for choosing the regularization term, i.e., for graphical model selection, without tuning was proposed. In Chapter 2 of this thesis, we establish a theoretical connection between the confidence level of graphical model selection via the DRO formulation and the asymptotic family-wise error rate of estimating false edges. Simulation experiments and real data analyses illustrate the utility of the asymptotic family-wise error rate control behavior even in finite samples.\par Next, we propose a completely tuning-free approach to estimating sparse precision matrix based on linear regression in Chapter 3. Theoretically, the proposed estimator is minimax optimal under various norms. In addition, we propose a second-stage enhancement with non-convex penalties, which possesses strong oracle properties. We assessed our proposed methods through comprehensive simulations and real data application on human gene network analysis