统计与数据科学学院知识库

缺失数据现象在观测数据中是十分常见的问题，而EM算法是针对缺失数据问题中求参数估计的常用方法，是将不完全数据转化为完全数据问题来处理。针对分层线性回归模型中参数估计问题，常常以两层数据模型为例，使用EM算法迭代求解。本文针对缺失数据的分层线性回归模型提出了Guass-Seidel型迭代方法，其主要思想是，迭代产生当前最新值，并运用最新值计算得到下一步最新参数估计值，对分层线性模型中固定效应与随机效应部分的参数估计进行推导，依据判断准则决定迭代过程的收敛，直至迭代过程结束。本文主要贡献之一在于利用Guass-Seidel型迭代算法提高参数迭代收敛速度。

对于处理统计推断中假设检验问题，统计推断是通过样本数据的观测信息来推断总体的主要方法，对于具有嵌套结构的分层线性回归模型的系数向量诊断方法，对分层线性回归模型的第一层模型系数诊断主要利用传统的线性嵌套回归模型检验进行统计推断。本文在对分层线性回归模型的第二层系数进行统计诊断中，将具有嵌套结构的多元线性回归模型推广到具有嵌套结构的分层线性回归模型中，主要构建分层线性回归模型似然函数比值来构造检验统计量，来判断第二层系数向量的拟合问题。

最后通过随机模拟的数值结果表明，与文献中的分层线性模型的EM算法相比，Guass-Seidel型迭代方法具有更有效的收敛速度。通过高校数学成绩数据来说明具有嵌套结构的分层线性模型的统计推断方法的有效性和实用性。

Missing data phenomenon is a very common problem in observation data, and EM algorithm is a common method for parameter estimation in missing data problem, which transforms incomplete data into complete data problem to deal with. For parameter estimation problem in Hierarchical Linear Regression Model, we often take two-level data model as an example, and use EM algorithm to solve iteratively. The main idea of this thesis is to generate the current latest value by iteration, and use the latest value to calculate the next step of the latest parameter estimation. The parameter estimation of fixed effect and random effect in Hierarchical Linear Model is derived, and the convergence of the iterative process is determined according to the judgment criteria until the end of the iterative process. The main contribution of this thesis is as follows One is to use the Guass-Seidel type iterative algorithm to improve the convergence speed of parameter iteration.

To deal with the problem of hypothesis testing in statistical inference, statistical inference is the main method to infer the population through the observation information of sample data. For the coefficient vector diagnosis method of Hierarchical Linear Regression Model with nested structure, the traditional F-test of Linear Nested Regression Model is mainly used to diagnose the coefficient of the first level model of Hierarchical Linear Regression Model. In this thesis, in the statistical diagnosis of the second layer coefficient of Hierarchical Linear Regression Model, we extend the multiple Linear Regression Model with nested structure to the Hierarchical Linear Regression Model with nested structure. We mainly construct the likelihood function ratio of the Hierarchical Linear Regression Model to construct the test statistics to judge the fitting problem of the second layer coefficient vector.

Finally, the numerical results of stochastic simulation show that the Guass-Seidel type iterative method has more effective convergence rate than the EM algorithm of Hierarchical Linear Model in literature. The validity and practicability of the statistical inference method of Hierarchical Linear Model with nested structure are illustrated by the data of college mathematics scores.