Bayesian Analysis Of Quantile Regression Under Complex Data

Compared with traditional mean regression model,quantile regression model can not only effectively deal with the statistical inference problem with outliers,but also has robustness for different distributions of observed data.With the wide application of quantile regression in finance,e conomy,b iomedicine a nd o ther fi elds,th e research on its statistical inference has become more and more in-depth.From the existing literature,we find t h at t h e r e search o n q u antile r e gression i s m o re c o nducted under the frequency framework,while the research based on Bayesian method is relatively less.Bayesian method attaches importance to the collection,mining and processing of historical and empirical information to form a priori,so as to improve the inference effect.It is worth pointing out that the existing researches on quantile regression based on Bayesian method mostly focus on algorithms and simulation analysis,and mainly consider linear quantile regression under complete data.In this thesis,based on Bayesian method,we study the Bayesian estimation of parameters and variable selection of semi-parametric quantile regression under missing data and longitudinal data,and theoretically discuss its asymptotic properties,including the construction of parameter posterior distribution.The likelihood functions are selected as empirical likelihood,asymmetric Laplacian likelihood and generalized moment quasi-likelihood,and the prior distributions are chosen as fixed p r iors and hierarchical priors.Specific research includes the following aspects:(1)In the case of missing at random for response variables or/and partial covariates,we study the Bayesian estimation and variable selection of parameters in partially linear varying-coefficient quantile regression model;Based on the local linear approximation for nonparametric functions,the empirical likelihood of linear parameters is constructed;Under the fixed p rior a s sumption,a p osterior d istribution b ased o n empirical likelihood is established,and the asymptotic properties of the posterior mean estimation and its asymptotic relationship with the maximum empirical likelihood estimation are studied;Meanwhile,we construct a Bayesian hierarchical model with variable selection attribute by using spike and slab priors and introducing potential binary variables,and prove the large sample properties of variable selection;The feasibility and effectiveness of the proposed method are verified by numerical simulation and real data analysis.(2)For the single-index quantile regression model under missing observations,under the assumption that the first component of the index parameter is 1,the link function is approximated by B-spline basis functions through the inverse probability weighted method,and the asymmetric Laplace likelihood of the index parameters is constructed using the profile principle;The Bayesian model and hierarchical model for parameters are given respectively under the fixed prior,the spike and slab priors;We study the asymptotic normality of Bayesian estimation for parameter,and establish the asymptotic relationship with the corresponding frequency estimation;Simultaneously,the results of variable selection based on hierarchical model are analyzed theoretically;In the simulation and empirical analysis part,the finite sample performance of the proposed method is studied.(3)For the partially linear single-index quantile regression model under longitudinal data,considering the correlation within individuals,the generalized moment quasi-likelihood function of the linear part and single-index part parameters is defined by using the spline basis function approximation link function technology;Under a fixed prior,we construct the posterior distribution of model parameters based on the generalized moment quasi-likelihood function,establish the asymptotic expansion of the posterior distribution and the asymptotic properties of the posterior mean estimation,and study the asymptotic relationship between the posterior distribution and the generalized moment frequency estimation;At the same time,the Bayesian variable selection model is established under hierarchical priors,and the large sample attributes of variable selection are discussed;Based on MCMC algorithm,the feasibility of the proposed method is studied through numerical simulation and actual data.The main contributions of this thesis are as follows.First,Bayesian analysis is carried out for quantile regression under missing data and longitudinal data to expand the application of Bayesian method in quantile regression;Second,for quantile regression,the analysis for the asymptotic properties of the parameter Bayesian estimation and its relationship with the corresponding frequency estimation provides theoretical support for the Bayesian quantile regression algorithm and statistical inference.Numerical simulation shows that if effective prior information is used,the estimation accuracy can be better improved,especially when the sample size is small;Thirdly,the theoretical analysis involved in Bayesian inference of these three semi-parametric quantile regression models has certain reference significance for the research of other semi-parametric Bayesian quantile regression models.