High Dimension Generalized Estimating Equation Estimation Consistency and Model Selection Consistency

We consider the problem of asymptotic theory and model selection for high dimensional Generalized Estimating Equation (GEE) on marginal regression analysis for clustered or longitudinal data. In this ``large $n$, diverging $p$" framework, we firstly establish the existence and consistency of the GEE estimator. Next we discuss the model selection and its consistency. As the GEE method only makes assumptions about the first two moments, the full likelihood is not specified. The likelihood based model selection criteria cannot be directly applied. This paper proposes two different information criteria. The first one applies simplified quasi-likelihood and the second one introduces a generalized model selection criterion based on a quadratic form of the residuals. Using the large deviation result of quadratic forms, we choose the appropriate penalty terms on the model complexity. The model selection consistency of the proposed criteria for divergent number of covariates is established for both simplified quasi-likelihood and the quadratic form of the residuals.