统计与数据科学学院知识库

       具有分层结构的数据在生活中是十分常见的，这类数据使用范围广，常用于 增长性研究、机构效应和综合研究。为了处理分层数据提出了分层线性模型，该 模型有两个基本假定：①组间误差项独立同分布且均值为零、方差有界；②各层 模型均为线性。然而，在实际研究中，数据经常会存在异方差或者重尾尖峰的情 况，协变量和响应变量之间也经常存在非线性的关系，所以，上面假定不能满足 实际对数据分析的需要。另外，分层线性模型在处理数据时采用的是均值回归， 它在给定协变量时只能体现响应变量的平均变化情况，不能刻画出响应变量的整 体条件分布，那么能够很好地刻画响应变量整体条件分布的方法就是分位数回 归。为了更好地解释协变量与响应变量之间可能存在的非线性关系，非参数分位 数回归模型的结合为解决这一问题提供了很好的思路，因此将非参数回归理论和 分位回归理论结合到分层模型将会很好的处理这类问题。

       本文首先将非参数回归理论和分位回归理论结合到分层模型中，同时加入混合效应，建立分层非参数混合效应模型，并使用基于检验函数的非参数分位数回归对其参数进行估计，在估计的过程中遇到了核函数和窗宽选择的挑战。研究表明，当数据样本容量足够大，不论选取哪种核函数，都可以在一定的正则条件下， 保证估计量具有相合性这一最基本的要求，所以本文选择了高斯核函数，并通过 最优渐近理论给出了最优窗宽与均值窗宽的关系。同时由于估计结果很难给出解 析解，所以将分位数回归与 EM 算法结合形成 EQ 算法，并进行迭代达到了估计 的目的。其次对于参数估计量的渐近性质进行了理论推导。接着通过 Monte Carlo 模拟，比较了随机误差项服从不同分布下的模型参数估计结果，可以发现，在样本量增大的情况下，本文所提出的模型的估计量都基本收敛到一个固定值，说明 所用的估计方法在该模型下具有稳健性。最后通过实际数据说明了模型和方法的 有效性和实用性。

  Data with hierarchical structure is very common in daily life. This kind of data is widely used in growth studies, institutional effects and synthesis studies. In order to deal with hierarchical data, a hierarchical linear model is proposed, which has two basic assumptions: (1) the error terms between groups are independent and identically distributed with zero mean and bounded variance; (2)The models of each layer are linear. However, in actual studies, data often have heteroscedasticity or heavy tail spikes, and there is often a non-linear relationship between covariates and response variables. Therefore, the above assumption cannot meet the actual needs of data analysis. In addition, hierarchical linear model adopts mean regression in data processing, which can only reflect the average change of response variables when given covariables, but cannot depict the overall conditional distribution of response variables. Therefore, quantile regression is a good method to depict the overall conditional distribution of response variables. In order to better explain the possible nonlinear relationship between the explained variables and explanatory variables, the combination of non-parametric quantile regression model provides a good idea to solve this problem. Therefore, the combination of non-parametric regression theory and quantile regression theory into hierarchical model will be a good solution to this problem. 

  In this thesis, nonparametric regression theory and quantil regression theory are combined into the layered model, and the mixed effect is added to establish the layered nonparametric mixed effect model, and the nonparametric quantile regression based on the test function is used to estimate its parameters. In the process of estimation, the challenges of kernel function and window width selection are encountered. The research shows that when the data sample size is large enough, no matter which kernel function is selected, the consistency of the estimator can be guaranteed under certain regular conditions, so the Gaussian kernel function is selected in this thesis, and the relationship between the optimal window width and the mean window width is given by the optimal asymptotic theory. At the same time, since it is difficult to give analytical solutions to the estimation results, quantile regression is combined with EM algorithm to form EQ two steps, and the purpose of estimation is achieved through iteration. Secondly, the asymptotic properties of parameter estimators are deduced theoretically. Then, the estimation results of model parameters with different distributions of error terms are compared by Monte Carlo simulation. It can be found that the estimators proposed in this thesis converge to a true value when the sample size increases, which proves the robustness of the estimation method. Finally, the effectiveness and practicability of the method are illustrated by practical data