Statistical Methods for Model Selection and Model Comparison

在本提案中，我的目标是对如何为复杂和高维数据开发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，目前还缺乏一个可用于GEE高维模型选择的信息标准。在这个建议中，我的目标是建立一个信息标准，这是模型选择一致的高维参数。我们建议使用两个不同的目标函数。在项目A1中，我们将使用伪高斯李克图作为模型拟合的度量。在项目A2中，我们将使用边际准似然。对于每个准则，我们将研究其渐近行为，并获得目标函数的大偏差结果。我将设计适当的惩罚项，以实现模型选择的一致性，在存在不同数量的参数。**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设真实参数在参数空间内部的正则性条件。然而，这种规律性条件在许多设置中可能被违反。在项目B1中，我们提出了一个条件复合似然比测试的情况下，感兴趣的参数驻留在参数空间的边界。而不是执行Bartlett校正的复合似然比，我们提出了一个测试，这是有条件的参数空间的分区上的观测数据的投影。在项目B2中，我们建议制定一个模型选择标准，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。二次规划算法可以用来确定投影。我们将利用这一结果设计适当的惩罚项，以保证准则的选择一致性。** C）有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。当边和顶点之间存在对称约束时，将向图形添加颜色以反映此类约束。模型比较和选择可以通过计算贝叶斯因子来执行。在项目C1中，我们提出了一种双可逆跳MCMC算法来估计贝叶斯因子。这将使我们能够在两个竞争模型之间进行模型比较。将研究通过整个模型空间的有效搜索算法。在项目C2中，我们提出了一种基于复合似然的融合LASSO算法来估计有色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 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.