Model Checking Of Quantile Regression Under Complex Data Structure

With the rapid development of computer performance and communication tech-nology,we will encounter various complex and high-dimensional data in many fields such as industrial production,biomedicine,and modern econometrics.To dig out the information hidden behind the data,such as studying the influence of some factors on the variables we are interested in,we often build bridges between relevant vari-ables with various regression models and then make corresponding statistical infer-ences based on the assumed models.To facilitate the interpretation of the relevant models'analysis results,the assumed models need to be as simple as possible,which often requires the participation of a large amount of prior knowledge.Assume that peo-ple doubt the model's original assumption or that the initial model assumption itself is wrong.In such cases,statistical inferences based on the model are often unconvincing,so it is necessary to make a reasonable test of the hypothetical existing model before making further statistical inferences.Parametric quantile regression models are often used to estimate the conditional quantile of the response variable Y after giving the covariable X.Compared with the classical mean regression model,the quantile regression has weaker requirements on the distribution of errors.It can provide conditional distribution information of re-sponses at different quantile levels,which significantly deepens the understanding of data and promotes the broad application of the quantile model.At present,there have been a large number of studies on model checking under mean regression.However,it is difficult to extend these methods directly to quantile regression models.In the quantile regression framework,the corresponding test statis-tics are no longer constructed based on the model's residual but are tested based on the quantile regression loss function's subgradient function.We will see the analytical difficulties caused by this change in the following theoretical analysis.Simultaneously,it is also of great practical value to study how to enhance the efficiency of testing in high-dimensional data and reduce the negative impact of data sparsity on testing in the case of high-dimensional data.Besides,few scholars have studied the testing of high-dimensional quantile regression models with missing data.This thesis will carry out relevant research on these issues.The following is a brief introduction to the main content of each part of this thesis.The first chapter of this thesis is the introduction,including some background knowledge.Firstly,it briefly reviews the general methods of model checking and intro-duces the quantile regression model testing's research status with complete observation data and random absence of response variables.Then,the quantile regression model and its related coefficient estimation methods are introduced.Finally,considering the importance of sufficient dimension reduction method to the test method proposed in this paper,we present several easy-to-use model dimension reduction methods.In particu-lar,the complete case assisted recovery(CCAR)method when the response variable is missing at random is presented in detail.In the second chapter of this thesis,a lack-of-fit test method for testing parametric single-index quantile regression models is constructed based on the kernel smoothing method.To avoid the curse of dimensionality in multivariate nonparametric estimation and to fully utilize the information contained in the model,we employ a sufficient di-mension reduction technique to identify the corresponding dimensionally reduced sub-space and then construct our test statistic in this subspace.The test statistic constructed in this way is similar to a local smoothing method that contains only one-dimensional covariates.At different quantile levels,the proposed test method is consistent against any global alternative hypothesis and can detect local alternative hypotheses distinct from the null hypothesis at a fast rate that existing local smoothing tests can achieve only when the model is univariate.A new wild bootstrap method was applied to ap-proximate the critical values of the quantile regression model test.The effectiveness of the method is verified by simulation experiments and a real data application.In the third chapter of this thesis,we consider the lack-of-fit test of parametric single-index quantile models when the response variable is missing at random.The model's coefficients are estimated by an estimation method suitable for the quantile regression coefficients of the missing data.Simultaneously,an algorithm for calculat-ing the central quantile subspace is given for the multi-dimensional quantile regression model with response missing at random.Based on the central quantile regression sub-space,we constructed two dimension reduction adaptive-to-model test statistics that are suitable for randomly missing response variables to avoid the curse of dimensionality.Under the null hypothesis and local alternative hypothesis,the asymptotic properties of the test statistics are obtained.It is shown that the proposed testing methods are consistent and can detect local alternative hypothetical models converging to the null model at the rate of order?(n^-1/2h^-1/4).A consistent bootstrap method is proposed to determine the critical values,and its asymptotic properties are established.The sim-ulation results show that the proposed method is superior to the existing methods in terms of both empirical sizes and powers in the case of multi-dimensional and even high-dimensional covariables.The ACTG Protocol 175 dataset is analyzed to show the application of the testing procedures.This thesis's results and conclusions are summarized in the last part,and the future research directions are pointed out.This thesis focuses on the model checking of the parametric single-index quantile regression models.Its main innovations are as follows:Firstly,under the complete observation sample,combined with the dimensionality reduction structure carried by the data itself,a test statistic that can automatically adapt to different models is con-structed,which avoids the curse of dimensionality caused by local smoothing in high-dimensional data.Simultaneously,the limiting null distribution of the proposed test statistics is obtained,and the local alternative hypotheses that are different from the null hypothesis can be detected quickly.Secondly,when the response variables are missing at random,the methods for calculating the central quantile subspace and the quantile regression coefficients are given,and their large-sample properties under the local al-ternative hypothesis are studied.Two test statistics suitable for responding to random missing are constructed based on the central quantile subspace under the missing data,proving the two test statistics'limit distribution under different hypotheses.Thirdly,unlike the model checking methods under the background of mean regression,when the observed data contain outliers,the test statistics in this thesis still have a significant advantage in terms of the test's size and power performance.Simultaneously,when the errors no longer satisfy the same distribution assumption,the simulation results also verify the proposed methods' robustness.