Quantile Regression For Hierarchical Nonparametria With Mixed Effects Model

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 quantile 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.