  We propose a new family of error distributions for model-based quantile
regression, 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. It thus overcomes the severe limitation of the asymmetric
Laplace distribution -- the most commonly used error model for parametric
quantile regression -- for which the skewness of the error density is fully
specified when a particular percentile is fixed. 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. Posterior
inference is implemented via Markov Chain Monte Carlo methods. The flexibility
of the new model relative to the asymmetric Laplace distribution is studied
through relevant model properties, and through a simulation experiment to
compare the two error distributions in regularized quantile regression.
Moreover, model performance in linear quantile regression, regularized quantile
regression, and Tobit quantile regression is illustrated with data examples
that have been previously considered in the literature.
