Statistical Methods for Model Selection and Model Comparison

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In this proposal, I aim to conduct theoretical investigations on how to develop model selection methods for complex and high dimensional data. A) Model Selection on High Dimensional Generalized Estimating Equations For clustered and longitudinal data, generalized estimating equations (GEE) have been widely used to perform parameter estimation. However, there is a lack of information criterion available for high dimensional model selection on GEEs. In this proposal, I aim to construct an information criterion which is model selection consistent with high dimensional parameters. We propose to work with two different objective functions. In project A1, we will use the pseudo Gaussian Likelihood as the measure of model fitting. In project A2, we will use the marginal quasi-likelihood. For each criterion, we will investigate its asymptotic behavior and obtain the large deviation result for the objective function. I will design the appropriate penalty term to achieve the model selection consistency in the presence of a divergent number of parameters. B) Statistical Inference and Model Selection Under Non-Standard Conditions Standard statistical theory often assumes the regularity condition that the true parameter is in the interior of the parameter space. However, such regularity condition can be violated in many settings. In project B1, we propose to develop a conditional composite likelihood ratio test for the situation where the parameters of interest reside on the boundary of the parameter space. Instead of performing Bartlett correction on the composite likelihood ratio, we propose a test which is conditional on the projection of the observed data on the partition of the parameter space. In project B2 we propose to develop a model selection criterion where the competing models have boundary constraints. We propose to project the log-likelihood ratio onto all the relative interiors of the tangent cone approximation of the parameter space. Quadratic programming algorithm can be used to determine the projections. We will use this result to design appropriate penalty term to ensure the selection consistency of the criterion. C) Model Selection for Colored Gaussian Graphical Models Gaussian graphical models (GGM) are used to describe network structures and relationships among the variables. When there are symmetry constraints among the edges and vertices, colors are added to the graph to reflect such constraints. Model comparison and selection can be performed by computing the Bayes factors. In Project C1, we propose to develop a double reversible jump MCMC algorithm to estimate the Bayes factors. This will enable us to make model comparisons between two competing models. Efficient search algorithm through the whole model space will be investigated. In Project C2, we propose to develop a composite likelihood based fused LASSO algorithm for the estimation of colored GGM.