On Bayesian Analysis Of Quantile Penalized Regression And Smoothing Spline

As the powerful supplement of mean regression,quantile regression enjoys the virtue of robustness to the outliers,no assumption of the distribution of random error.It is firstly proposed by Koenker and Bassett(1978),then there are more and more literature on its theory and applications in the last decades(Koenker et al.,1994;Koenker and Machado,1999;Liu and Wu,2011;Wang et al.,2013).Bayesian analysis,due to its virtue of small samples performance and automatic uncertainty evaluation of parameters,plays an important role in the theory and application of quantile analysis.Nowadays,There has been a lot of researches for quantile regression under the Bayesian framework(Li et al.,2010;Thompson et al.,2010;Kottas and Krnjajic 2009;Reich et al.,2011;Rodrigues and Fan2017;Rodrigues et al.,2019).This thesis focuses on three different areas in quantile regresion topic under Bayesian framework,Bayesian penalized quantile regression,Bayesian quantile smoothing spline and Bayesian composite quantile smoothing spline curve fitting.For Bayesian penalized quantile regression and Bayesian quantile smoothing spline,we point out the undesirable issues in the current literature and we develop new methodology to overcome the drawbacks.For Bayesian composite quantile smoothing spline curve fitting,based on whether the underlying quantile curve are parallel or not,we propose two Bayesian methods to fit the multiple quantile curves simultaneously.In addition,our methods can keep the fitted curves non-crossing.Our major works are listed as below.(1)We systematically investigate the choice of prior and computation problem of Bayesian quantile LASSO and its generalization.We point out that there are several undesirable issues in the current literature(Li et al.,2010;Alhamzawi and Algamal,2019;Alhamzawi et al.,2019)as follows:(i)the joint posterior could be multimodal;(ii)the estimation of posterior may be sensitive to the choice of hyperparameters;(iii)the objective prior for the scale parameter may be problematic.We propose a new prior of the regression coefficient to alleviate the issues in the current literature.We propose the sufficient and necessary conditions of the posterior propriety and show that our methods still work for the large p small n case while the existing methods could not apply.In addition,we propose the partially collasped Gibbs sampling to make the sampling procedure more efficient.(2)We research on the prior choice,computation of posterior and the generalization of theory of Bayesian quantile smoothing splines.Bayesian quantile smoothing spline is firstly investigated by Thompson et al.,(2010).However,they chose the fixed scale parameter and it may lead to the bad performance when the quantile is away from 0.5(Santos and Bolfarine,2016).In addition,we find the estimation of smoothing parameter is not stable using the method proposed by Thompson et al.,(2010),which may lead to a flat or a over-fitting quantile curve.To resolve this problem,we systematically study the objective choice on the scale parameter and obtain the condition of posterior propriety.In addition,we extend our theoretical result to quantile smoothing spline with the unobserved knots.Simulation studies and real example show our method outperforms the competing methods.(3)We develop two new Bayesian approaches for composite quantile curve fitting.Single quantile curve fitting method may have the crossing problem for the estimated quantile curves.The crossing quantile fitted curves disobey the basic fact that quantile function is nondecreasing function of quantile.That the estimated quantile curves cross with each other is called 'crossing problem',which is the popular topic in the quantile regression.A lot of literature has tried to alleviate this crossing problem(He,1997;Muggeo et al.,2013;Chernozhukov,2005;Dette and Volgushev 2008;Rodrigues and Fan,2017;Rodrigues et al.,2019).We propose two novel models to deal with the quantile crossing problem.For the first one,we assume the underlying quantile curves are parallel;For the second one,we assume the underlying quantile curves are non-parallel.We propose the new minimization problem to keep the estimated quantile curves non-crossing,then in order to handle the minimization problem under the Bayesian framework,we proposed the corresponding prior,pseudo likelihood and the Gibbs sampling algorithm We employ the Montle Carlo expectation-maximization algorithm to pick the penalty parameters.Simulation results show that our methods have the better performances compared with the current quantile curve fitting methods.