| We propose semiparmetric Bayesian quantile regression methods for analyzing independent and clustered data and Bayesian regularized quantile regression methods for quantile and interquantile shrinkage.;Bayesian quantile regression is challenging without making any parametric likelihood assumptions. Instead of posing any parametric distributional assumptions on the random errors, we approximate the central density by linearly interpolating the conditional quantile functions of the response at multiple quantiles and estimate the tail densities by adopting extreme value theory.;In Chapter 2, we propose a semiparametric Bayesian quantile regression method for analyzing independent data based on the proposed approximate likelihood. In Chapter 3, we develop a semiparametric Bayesian quantile regression method for analyzing clustered data, where random effects are included to accommodate the intra-cluster dependence. Through joint-quantile modeling, the proposed methods yield the joint posterior distribution of quantile coefficients at multiple quantiles and meanwhile avoid the quantile crossing issue. Through simulation studies, we demonstrate that the proposed methods lead to more efficient multiple-quantile estimation than existing methods for quantile regression in finite samples.;In Chapter 4, we propose penalization methods for quantile and interquantile shrinkage for linear quantile regression. Our proposed methods shrink quantile slopes and their interquantile differences towards zero in some or the entire quantile region by using fused adaptive LASSO or group LASSO penalties based on Laplace priors. Our numerical investigations show that the proposed methods yield higher estimation efficiency than the conventional quantile regression method when there exists sparsity in the quantile coefficients and/or interquantile differences. |
