This paper investigates the property of the penalized estimating equations when both the mean and association structures are modelled. To select variables for the mean and association structures sequentially, we propose a hierarchical penalized gener
alized estimating equations (HPGEE2) approach. The first set of penalized estimating equations is solved for the selection of significant mean parameters. Conditional on the selected mean model, the second set of penalized estimating equations is solved for the selection of significant association parameters. The hierarchical approach is designed to accommodate possible model constraints relating the inclusion of covariates into the mean and the association models. This two-step penalization strategy enjoys a compelling advantage of easing computational burdens compared to solving the two sets of penalized equations simultaneously. HPGEE2 with a smoothly clipped absolute deviation (SCAD) penalty is shown to have the oracle property for the mean and association models. The asymptotic behavior of the penalized estimator under this hierarchical approach is established. An efficient two-stage penalized weighted least square algorithm is developed to implement the proposed method. The empirical performance of the proposed HPGEE2 is demonstrated through Monte-Carlo studies and the analysis of a clinical data set.
In this article, we discuss the composite likelihood estimation of sparse Gaussian graphical models. When there are symmetry constraints on the concentration matrix or partial correlation matrix, the likelihood estimation can be computational intensi
ve. The composite likelihood offers an alternative formulation of the objective function and yields consistent estimators. When a sparse model is considered, the penalized composite likelihood estimation can yield estimates satisfying both the symmetry and sparsity constraints and possess ORACLE property. Application of the proposed method is demonstrated through simulation studies and a network analysis of a biological data set.
In a Gaussian graphical model, the conditional independence between two variables are characterized by the corresponding zero entries in the inverse covariance matrix. Maximum likelihood method using the smoothly clipped absolute deviation (SCAD) pen
alty (Fan and Li, 2001) and the adaptive LASSO penalty (Zou, 2006) have been proposed in literature. In this article, we establish the result that using Bayesian information criterion (BIC) to select the tuning parameter in penalized likelihood estimation with both types of penalties can lead to consistent graphical model selection. We compare the empirical performance of BIC with cross validation method and demonstrate the advantageous performance of BIC criterion for tuning parameter selection through simulation studies.
