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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 regre

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 regre