Model Specification Tests Of Quantile Regression Models

With the continuous development and application of statistics,the mean regression model has received a lot of attention due to its wide applicability since it was proposed.The quantile regression model,as an extension of the mean regression model,studies the relationship between the predictor variable and the conditional quantile of the response variable.Compared with the mean regression model,the quantile regression model is more robust and can obtain information about the distribution of response variables from multiple perspectives,which has become a hotspot in recent years.With the deepening of research and the continuous improvement of demand,the application field of quantile regression method has gradually expanded,and the research on models based on quantile regression has become more and more abundant.However,there are still some interesting things in the field of quantile regression.The problem of statistical inference of the model of the Ordinary quantile regression at the quantile level has certain limitations,and the results are often affected by the selection of a single quantile point,so we further study the statistical inference problem of the quantile process regression model.With the improvement of data collection and analysis level by technicians,we find that more and more data presents functional characteristics,so we also discuss the functional linear quantile regression model of our interest in the last subsection of the article.Model specification testing problem.In our first chapter,we briefly review the quantile regression literature,and give an introduction to those reasons why we need to consider such statistical inferences under quantile regression.In addition,we also introduce the partial linear additive quantile model that will be considered in the following chapter.At the same time,we know that it might not be sufficient to focus on a single quantile level,so quantile regression process is important,which is introduced in Chapter 1.In addition to this,we briefly describe the reasons why we focus on functional regression models.In order to facilitate the subsequent research,in the second chapter of the dissertation we give more details on those models and related methods which will be used in the following chapters,including some properties of the B-spline basis function,the basic assumptions of the quantile regression models,and related the asymptotic results of the estimators.In Chapter 3,the model specifications are considered under linear additive quantile regression models.We focuses on the specification with polynomial models for quantile regression.We first propose a two-step estimation method based on B-splines and give the asymptotic properties of the corresponding estimator;and then we study the relationship between the polynomial form and the corresponding non-parametric components of the model,which leads to a novel test.Our simulation study shows the superior performance of the test.We then demonstrate the usefulness of the proposed method with a dataset of China’s urban population,which explores the potential relationship between the age and the income and gives results that are quite consistent with existing social studies.In Chapter 4,we study the quantile process regression model.First,under the quantile process regression model,we use B-splines to approximate the quantile coefficient process,and obtain the parameter estimates with the MM algorithm.Subsequently,we establish the asymptotic results of the coefficient process estimator.In addition,we also propose a lack-of-fit test method for parametric model specifications of the quantile process regression model.The lack-of-fit test statistics are proposed by adopting the spirit of a non-negative statistic used in the two-sample test.Through simulation experiments,we demonstrate the validity of our proposed method.We also applied this method to Boston housing price data for analysis,and obtained reasonable conclusions.In Chapter 5,we focus on the functional linear quantile regression model.We mainly study the model specification test under this model,that is,given a quantile level,we will check whether the assumption that the response variable has a linear relationship with the functional explanatory variable is correct or not.We use B-splines to approximate the coefficients of the functional linear model and propose a lack-of-fit testing method.Through Monte Carlo simulation studies and real data analysis for meat Spectral Data,it is verified that the proposed method has good performance in the finite-sample cases.Finally,we summarize the main contributions of this work,comment on those proposed methods,and discussed the possible future work.