Bayesian Statistical Inference For Quantile Regression Models

Quantile regression(QR)modeling is an important and widely used regression method,and it can be regarded as an significant complement to mean regression method.Due to its many excellent merits,QR is receiving more and more extensive attention and is widely used in economic,financial,data mining,environmental and other fields.For QR estimation,the commonly used algorithms are simplex algorithm,interior point method,and smoothing method.While,under Bayesian framework,one should specify a reasonable distribution,and the commonly used methods are substitution likelihood method,Bayesian empirical likelihood and Bayesian generalized method of moment.In recent years,more and more researchers consider QR under Bayesian framework,because it not only has the merits of QR,but also shares the advantage of using auxiliary information such as historical data and empirical knowledge to improve inference.In this thesis,we propose using Bayesian QR framework for parameter estimation and variable selection in linear model with non-ignorable missing covariates,structural equation models,partially linear single-index model are considered,respectively.For linear model with non-ignorable missing covariates,the problem of parameters estimation,variable selection and local influence analysis are considered.For structural equation models,robust estimation based on composite QR method is obtained,and the latent variable selection with different methods is discussed.For partial linear single-index model with non-ignorable missing response,parameters estimation and variable selection is considered,and case-deletion influence analysis is discussed.The main contents of this thesis are summarized as follows:Firstly,Bayesian inference on QR model with mixed discrete and non-ginorable missing covariates is considered.The probit regression model is used to specify the missing data mechanism.The missing covariates are assumed to be exponential family distributions,and the response variable is approximated by asymmetric laplacedistribution.A hierarchical structure is employed to reformulate the considered QR model.A hybrid algorithm combining the Gibbs sampler and the Metropolis-Hastings algorithm is developed to simultaneously produce Bayesian estimates of unknown parameters and latent variables as well as their corresponding standard errors.A Bayesian variable selection method is also proposed to simultaneously obtain Bayesian estimates of unknown parameters and recognize significant covariates.A Bayesian local influence approach is presented to assess the effect of minor perturbations to the data,priors and sampling distributions on posterior quantities of interest.Secondly,for structural equation models,composite QR method is considered to obtain robust estimation.Mixed normal distribution are assigned to explanatory variables,and manifest variables and outcome latent variable are assumed to be pseudo composite ALDs to conduct composite QR estimation.Based on the idea of penalized likelihood,the effect of simultaneous parameter estimation and variable selection is achieved by penalizing the regression coefficients in the structural equation model and transforming the penalty term into a prior of the parameters under the Bayesian framework.Various variable selection methods,including Bayesian Lasso,Bayesian Adaptive Lasso,Bayesian Elastic Net and Bayesian Grouped Lasso,are used to select significant explanatory latent variables,and the effectiveness of these methods are discussed.Lastly,for partial linear quantile single-index model with non-ignorable missing response,the response variable is modeled by finite mixture ALD,the non-ignorable missing mechanism is fitted by a logistic regression model,and the single index link function is analyzed by considering Gaussian process prior,while the uniform distribution and double index distribution on the unit sphere are considered as prior of the index vector respectively.The posterior distribution for the unknown parameters are derived,and the posterior inference of the proposed approach is performed via MCMC method.The diagnosis of our proposed model is discussed by Bayesian case-deletion.