The focus of this work is to develop a Bayesian framework to combine information from multiple parts of the response distribution characterized with different quantiles. The goal is to obtain a synthesized estimate of the covariate effects on the response variable as well as to identify the more influential predictors. This framework naturally relates to the traditional quantile regression, which studies the relationship between the covariates and the conditional quantile of the response variable and serves as an attractive alternative to the more widely used mean regression methods. We achieve the objectives through constructing a Bayesian mixture model using quantile regressions as the mixture components.

The first stage of the research involves the development of a parametric family of distributions to provide the mixture kernel for the Bayesian quantile mixture regression. We derive a new family of error distributions for model-based quantile regression called generalized asymmetric Laplace distribution, which is constructed through a structured mixture of normal distributions. The construction enables fixing specific percentiles of the distribution while, at the same time, allowing for varying mode, skewness and tail behavior. This family provides a practically important extension of the asymmetric Laplace distribution, which is the standard error distribution for parametric quantile regression. We develop a Bayesian formulation for the proposed quantile regression model, including conditional lasso regularized quantile regression based on a hierarchical Laplace prior for the regression coefficients, and a Tobit quantile regression model.

Next, we develop the main framework to model the conditional distribution of the response with a weighted mixture of quantile regression components. We specify a common regression coefficient vector for all components to synthesize information from multiple parts of the response distribution, each modeled with one quantile regression component. The goal is to obtain a combined estimate of the predictive effect of each covariate. We consider the following two choices of kernel densities for the mixture model. When the probability of the quantile in each regression component is known, we model the components with the generalized asymmetric Laplace distribution, as its shape parameter introduces flexibility in shape and skewness to the kernel; else when the quantile probabilities are unknown, we use the asymmetric Laplace distribution as kernel density and view its skewness parameter, which is also the quantile probability of the component, as a random quantity and estimate it from the data. Under each kernel density, we formulate the hierarchical structure of the mixture weights and develop the approach to the posterior inference. We consider both parametric and nonparametric priors for the framework, and explore inferences for the number of components to be included. We demonstrate the performance of the method in identification of influential variables with simulation examples and illustrate the posterior predictive inferences in a realty price data from the Boston metropolitan area.

Finally, we extend the framework to apply the methods to specific problems in survival analysis and epidemiology. Both applications involve analyses of two cohorts, which oftentimes exhibit differing responses given the same predictor input. We adapt the proposed framework to model the survival data with right-censoring. For applications in epidemiology, we study the ordering properties of the mixture kernels and incorporate stochastic ordering in the two-cohort mixture framework through structured priors, which conforms with the assumption in certain circumstances of receiver operating characteristic curve estimation. With the adapted models, we carry out cohort-specific identification of influential variables and gain insights into the contribution in estimation and prediction from different parts of the response distribution, which are depicted by the corresponding quantile regression components. We illustrate the applications with a time-to-event data set on length of stay at nursing home and two disease diagnosis data sets, one on adolescent depression and the other on cattle epidemics.