Statistical Methods for Model Selection and Model Comparison

在这个提案中，我的目标是对如何开发复杂高维数据的模型选择方法进行理论研究。A)高维广义估计方程的模型选择对于聚类和纵向数据，广义估计方程（GEE）已被广泛用于进行参数估计。然而，目前还没有足够的信息标准来进行高维模型的选择。在这个方案中，我的目标是构建一个信息准则，即与高维参数一致的模型选择。我们建议使用两个不同的目标函数。在项目A1中，我们将使用伪高斯似然作为模型拟合的度量。在项目A2中，我们将使用边际拟似然。对于每个准则，我们将研究其渐近行为，并得到目标函数的大偏差结果。我将设计适当的惩罚项，以在参数数量分散的情况下实现模型选择的一致性。B)非标准条件下的统计推断和模型选择标准统计理论通常假定真参数位于参数空间内部的正则性条件。然而，在许多情况下，这种规则条件可能被违反。在项目B1中，我们建议针对感兴趣的参数位于参数空间边界的情况开发条件复合似然比检验。我们提出了一种以观测数据在参数空间分区上的投影为条件的检验，而不是对复合似然比进行Bartlett校正。在项目B2中，我们建议开发一个模型选择标准，其中竞争模型具有边界约束。我们建议将对数似然比投影到参数空间的切锥近似的所有相对内部。二次规划算法可用于确定投影。c)彩色高斯图模型的模型选择高斯图模型（GGM）用于描述网络结构和变量之间的关系。当边缘和顶点之间存在对称约束时，在图形中添加颜色以反映这种约束。模型的比较和选择可以通过计算贝叶斯因子来完成。在C1项目中，我们提出了一种双可逆跳跃MCMC算法来估计贝叶斯因子。这将使我们能够在两个相互竞争的模型之间进行模型比较。研究整个模型空间的高效搜索算法。在C2项目中，我们提出了一种基于复合似然的融合LASSO算法来估计彩色GGM。

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 EquationsFor 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 ConditionsStandard 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 ModelsGaussian 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.